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Discrete Mathematics 342 (2019) 2092–2099
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On minimally 2-(edge)-connected graphs with extremal spectral radius ∗
Xiaodan Chen a , Litao Guo b , a b
College of Mathematics and Information Science, Guangxi University, Nanning 530004, Guangxi, PR China School of Applied Mathematics, Xiamen University of Technology, Xiamen 361024, Fujian, PR China
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Article history: Received 7 September 2018 Received in revised form 24 January 2019 Accepted 1 April 2019 Available online 30 April 2019
a b s t r a c t A graph is minimally 2-(edge)-connected if it is 2-(edge)-connected and deleting any arbitrary chosen edge always leaves a graph which is not 2-(edge)-connected. In this paper, we completely characterize the minimally 2-(edge)-connected graphs having the largest and the smallest spectral radius, respectively. © 2019 Elsevier B.V. All rights reserved.
Keywords: Minimally 2-connected graph Minimally 2-edge-connected graph Spectral radius Extremal graph
1. Introduction Throughout this paper we consider finite, undirected and simple graphs. Let G be a graph with vertex set V (G) = {v1 , v2 , . . . , vn } and edge set E(G). For i ∈ {1, 2, . . . , n}, denote by NG (vi ) and dG (vi ) the neighborhood and the degree of the vertex vi in G, respectively. A graph G is said to be r-regular if dG (vi ) = r for each i ∈ {1, 2, . . . , n}. We also write ∆(G) and δ (G) for the maximum degree and the minimum degree of G, respectively. As usual, let Kn , Cn and Pn denote the complete graph, the cycle and the path on n vertices, respectively. Let Ka, b , also, denote the complete bipartite graph with two partite sets having a and b vertices, respectively. The adjacency matrix of a graph G is defined as the n × n square matrix A(G) = (aij ) whose entries are given by
{ aij =
1 0
if vi vj ∈ E(G), otherwise.
The spectral radius of G, denoted by ρ (G), is defined to be the spectral radius of A(G), i.e., the largest eigenvalue of A(G). The investigation on the spectral radius of graphs, which could date back to the seminal work of Collatz and Sinogowitz [8] in the 1950s, has been a central research theme in spectral graph theory. There have been numerous results concerning the spectral radius of graphs; for details one may refer to a survey by Cvetković and Rowlinson [10] in 1990 and a recent research monograph by Stevanović [24]. It is noted that most of these results relate to the famous Brualdi– Solheid’s general question [4] that asks to characterize graphs with extremal values of the spectral radius in a given class of graphs (where ‘‘extremal’’ usually means ‘‘maximal’’); see [1,2,6,7,11,12,15–23,25–27,29,30] for more information. We here mention a classical result of this type. ∗ Corresponding author. E-mail addresses: [email protected] (X. Chen), [email protected] (L. Guo). https://doi.org/10.1016/j.disc.2019.04.002 0012-365X/© 2019 Elsevier B.V. All rights reserved.
X. Chen and L. Guo / Discrete Mathematics 342 (2019) 2092–2099
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Fig. 1. The graphs Fq , Fq∗ , Fp1, q and Fp2, q .
Theorem 1.1 (See [14]). If T is a tree of order n ≥ 2, then 2 cos
π n+1
≤ ρ (T ) ≤
√
n − 1,
with equality in the left (resp., right) inequality if and only if T ∼ = Pn (resp., T ∼ = K1, n−1 ). A graph is k-connected if removing fewer than k vertices always leaves the remaining graph connected, and is minimally k-connected if it is k-connected and deleting any arbitrary chosen edge always leaves a graph which is not k-connected. Analogously, a graph is k-edge-connected if removing fewer than k edges always leaves the remaining graph connected, and is minimally k-edge-connected if it is k-edge-connected and deleting any arbitrary chosen edge always leaves a graph which is not k-edge-connected. Observe that a graph is minimally 1-(edge)-connected if and only if it is a tree. So, it is natural to ask which graphs have the extremal spectral radius among all minimally k-(edge)-connected graphs for k ≥ 2. In this paper, we are concerned with the extremal spectral radius of minimally 2-(edge)-connected graphs and obtain the following two results. Theorem 1.2.
If G is a minimally 2-connected graph of order n ≥ 4, then
2 ≤ ρ (G) ≤
√
2n − 4,
with equality in the left (resp., right) inequality if and only if G ∼ = Cn (resp., G ∼ = K2, n−2 ). Theorem 1.3. Let G be a minimally 2-edge-connected graph of order n ≥ 4. (i) ρ (G) ≥ 2, with equality if and only if G ∼ = Cn . √ (ii) If n ̸ = 5, then ρ (G) ≤ 2n − 4, with equality if and only if G ∼ = K2, n−2 ; if n = 5, then ρ (G) ≤ and only if G ∼ = F2 , where F2 is the friendship graph having two triangles (see Fig. 1).
√ 1+ 17 , 2
with equality if
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The proofs of Theorems 1.2 and 1.3 will be presented in the forthcoming two sections, and a brief remark will be given in the final section. 2. Proof of Theorem 1.2 In order to give the proof of Theorem 1.2, we need some auxiliary results. The first one is well known, which can be found in [9], pp. 84–85. Lemma 2.1 (See [9]). For any graph G, we have ρ (G) ≥ δ (G), with equality if and only if G is regular. The next result comes from Favaron, Mahéo and Saclé [14]. Lemma 2.2 (See [14]). If G is a connected graph, then
ρ (G) ≤ max
v∈V (G)
√ ∑
dG (u).
u∈NG (v )
Equality holds if and only if G is regular or bipartite semiregular. For X , Y ⊂ V (G), we denote by e(X ) the number of edges in the subgraph induced by X and by e(X , Y ) the number of edges with one endpoint in X and one endpoint in Y . It is easy to check that (see also [27])
∑
dG (u) = 2e(NG (v )) + e(NG (v ), V (G)\NG (v )).
(2.1)
u∈NG (v )
We also need some classical results concerning the structure of minimally 2-connected graphs. Lemma 2.3 (See [3], p. 211). If G is a minimally 2-connected graph, then δ (G) = 2. Lemma 2.4 (See [13]). A minimally 2-connected graph with more than three vertices contains no triangles. Lemma 2.5 (See [13]). A minimally 2-connected graph G with n ≥ 4 vertices has as most 2n − 4 edges, with equality if and only if G ∼ = K2, n−2 . The following result is a direct consequence of Lemma 2.5. Lemma 2.6.
Let G be a minimally 2-connected graph of order n ≥ 4. Then for any vertex v ∈ V (G),
e(NG (v ), V (G)\NG (v )) ≤ 2n − 4, with equality if and only if G ∼ = K2, n−2 . We are now ready to give the proof of Theorem 1.2. Proof of Theorem 1.2. Let G be a minimally 2-connected graph of order n ≥ 4. Clearly, by Lemmas 2.1 and 2.3, we have ρ (G) ≥ δ (G) = 2, with equality if and only if G is 2-regular, that is, G ∼ =√Cn . 2n − 4 when G ≇ K2, n−2 , because ρ (K2, n−2 ) = For the upper bound on ρ (G), we just need to show that ρ (G) < √ 2n − 4 (see [9], p. 72). Suppose now that G ≇ K2, n−2 . For any vertex v ∈ V (G), by Lemma 2.6, we obtain e(NG (v ), V (G)\NG (v )) <√2n − 4. Moreover, by Lemma 2.4, we have e(NG (v )) = 0. Consequently, from (2.1) and Lemma 2.2 it follows that ρ (G) < 2n − 4, as desired. This completes the proof of Theorem 1.2. □ 3. Proof of Theorem 1.3 For proving Theorem 1.3, we need two necessary results. Lemma 3.1 (See [3], p. 219). If G is a minimally 2-edge-connected graph, then δ (G) = 2. For q ≥ 1, let Fq be the friendship graph of order n = 2q + 1, that is, the graph consisting of q triangle(s) intersecting in a common vertex. Let Fq∗ be the graph of order n = 2q + 1 (q ≥ 3) obtained from Fq−1 by coalescing one of its noncentral vertices and a vertex of K3 . Also, for p ≥ 2 and q ≥ 1, let Fp1, q (resp., Fp2, q ) be the graph of order n = p + 2q + 2 obtained from Fq by coalescing its center x and a p-degree (resp., 2-degree) vertex of K2, p . Fig. 1 shows the graphs Fq , Fq∗ , Fp1, q and Fp2, q ; obviously, all these graphs are minimally 2-edge-connected.
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Lemma 3.2.
2095
Let G be a minimally 2-edge-connected graph of order n ≥ 4. If G ≇ F(n−1)/2 , then for any vertex v ∈ V (G),
2e(NG (v )) + e(NG (v ), V (G)\NG (v )) ≤ 2n − 4.
(3.1) Fpλ, (n−p−2)/2
Moreover, the equality holds in (3.1) if and only if G ∼ (λ = 1, 2) and v = K2, n−2 (and v is any vertex of G), or G ∼ = is the unique cut vertex of G, or G ∼ = F(n∗ −1)/2 and v is the cut vertex of G whose degree is as large as possible. Proof. If G ∼ = K2, n−2 , then it can be directly checked that the equality holds in (3.1) for any vertex v ∈ V (G). It is also not difficult to verify that if G ∼ = Fpλ, (n−p−2)/2 (λ = 1, 2), then the equality holds in (3.1) only for the unique cut vertex of G, ∗ ∼ and that if G = F(n−1)/2 , then the equality holds in (3.1) only for the cut vertex of G whose degree is as large as possible. Therefore, to complete the proof, we just need to show that 2e(NG (v )) + e(NG (v ), V (G)\NG (v )) < 2n − 4
(3.2)
∗ holds for any vertex v ∈ V (G) when G ≇ F(n−1)/2 , K2, n−2 , Fpλ, (n−p−2)/2 (λ = 1, 2) and F(n −1)/2 . ∗ . We denote by σ the number of cut vertices Suppose now that G ≇ F(n−1)/2 , K2, n−2 , Fpλ, (n−p−2)/2 (λ = 1, 2) and F(n −1)/2 in G and consider the following two cases. Case 1. σ = 0. In this case, G is 2-connected. Furthermore, G is minimally 2-connected, because G is minimally 2-edge-connected. Thus, for any vertex v ∈ V (G), by Lemma 2.4, we have e(NG (v )) = 0. This, together with Lemma 2.6 and the assumption that G ≇ K2, n−2 , yields (3.2) immediately. Case 2. σ ≥ 1. Let w1 , . . . , wσ denote the cut vertices of G. For each i ∈ {1, . . . , σ }, let Bi1 , Bi2 , . . . , Bisi be all the blocks of G which contain wi ; note that si ≥ 2. Clearly, for each j ∈ {1, 2, . . . , si }, Bij is minimally 2-edge-connected, since G is so. Furthermore, because G is simple, we have |V (Bij )| = nij ≥ 3 and then, Bij is minimally 2-connected. We will say that a block Bij is type I if nij ≥ 4, and is type II if nij = 3. For any vertex v ∈ V (Bij ), if Bij is type I, then by Lemmas 2.4 and 2.6, we obtain
e(NBij (v )) = 0,
(3.3)
e(NBij (v ), V (Bij )\NBij (v )) ≤ 2nij − 4,
(3.4)
and
with equality if and only if Bij ∼ = K3 and consequently, we have = K2, nij −2 ; if Bij is type II, then it is easy to see that Bij ∼ e(NBij (v )) = 1,
(3.5)
e(NBij (v ), V (Bij )\NBij (v )) = 2.
(3.6)
and
Set Gi := Bi1 ∪ Bi2 ∪ · · · ∪ Bisi . Let ti be the number of blocks of type I in Gi and without loss of generality, suppose that Bi1 , Bi2 , . . . , Biti are such blocks. Clearly, 0 ≤ ti ≤ si and there are no blocks of type I in Gi if ti = 0. It is also easy to see that Gi is a (not necessary proper) subgraph of G with wi being its unique cut vertex and, ti ⏐ ⏐ ∑ ⏐V (Gi )⏐ = nij − (ti − 1) + 2(si − ti ).
(3.7)
j=1
Now, for any vertex v ∈ V (G), we denote by θ (v ) the number of the cut vertices of G adjacent to v , and divide into the following two subcases. Subcase 2.1. v is one of the cut vertices of G. Without loss of generality, we can suppose that v = wσ . Clearly, if σ = 1, then Gσ ∼ = G and hence, |V (Gσ )| = n; if σ ≥ 2, then
⏐ ⏐ ⏐V (Gσ )⏐ ≤ n − 2,
(3.8)
with equality if and only if G is the graph obtained by coalescing any non-cut vertex of Gσ and a vertex of K3 (in this case,
σ = 2). It is also observed that (⋃ ) ⋃( ⋃ ) tσ sσ NG (v ) = NGσ (v ) = NBσ j (v ) NBσ j (v ) , j=1
(3.9)
j=tσ +1
and NBσ i (v ) NBσ j (v ) = ∅ for any 1 ≤ i ̸ = j ≤ sσ . Since each edge of G only belongs to one block of G, by (3.9), (3.3) and (3.5), we have
⋂
e(NG (v )) = e(NGσ (v )) =
tσ ∑ j=1
e(NBσ j (v )) +
sσ ∑ j=tσ +1
e(NBσ j (v )) = sσ − tσ .
(3.10)
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Note that 0 ≤ θ (v ) ≤ σ − 1. If θ (v ) = 0, then tσ ≥ 1 (as G ≇ F(n−1)/2 and any cut vertex in a block of type II in Gσ is always adjacent to v ). Moreover, observing that NG (u) ⊆ V (Gσ ) for any vertex u ∈ NG (v ), we obtain e(NG (v ), V (G)\NG (v )) = e(NGσ (v ), V (Gσ )\NGσ (v )).
(3.11)
We further observe that NGσ (u) ⊆ V (Bσ j ) for any vertex u ∈ NBσ j (v ), which, together with (3.9), (3.4), (3.6) and (3.7), yields that e(NGσ (v ), V (Gσ )\NGσ (v ))
=
tσ ∑
e(NBσ j (v ), V (Bσ j )\NBσ j (v ))
j=tσ +1
j=1
≤
sσ ∑
e(NBσ j (v ), V (Bσ j )\NBσ j (v )) +
tσ ∑
(2nσ j − 4) + 2(sσ − tσ )
j=1
⏐ ⏐ = 2⏐V (Gσ )⏐ − 2(tσ + 1) − 2(sσ − tσ ),
(3.12)
with equality if and only if Bσ j ∼ = K2, nσ j −2 for all j ∈ {1, . . . , tσ }. We here remark that if σ = 1 and tσ = 1, then the strict inequality holds in (3.12), because G ≇ Fpλ, (n−p−2)/2 (λ = 1, 2). Now, from (3.10), (3.11) and (3.12), it follows that 2e(NG (v )) + e(NG (v ), V (G)\NG (v )) ≤ 2⏐V (Gσ )⏐ − 2(tσ + 1),
⏐
⏐
with the strict inequality if σ = 1 and tσ = 1. This, together with the fact that tσ ≥ 1 and |V (Gσ )| ≤ n with equality if and only if σ = 1, yields (3.2). If θ (v ) = θ ≥ 1, then σ ≥ 2. Moreover, we suppose, without loss of generality, that w1 , . . . , wk belong to some blocks of type I in Gσ , and wk+1 , . . . , wθ belong to some blocks of type II in Gσ ; note that 0 ≤ k ≤ θ and, w1 , . . . , wθ all belong to some blocks of type I in Gσ if k = θ , and all belong to some blocks of type II in Gσ if k = 0. Next, for each i ∈ {1, . . . , θ }, consider Gi and its blocks Bi1 , Bi2 , . . . , Bisi . Recalling that Bij is minimally-2-connected, if Bij is type I (in this case, 1 ≤ j ≤ ti ), then dBij (wi ) ≤ nij − 2,
(3.13)
and if Bij is type II (in this case, ti + 1 ≤ j ≤ si ), then dBij (wi ) = 2.
(3.14)
Moreover, for convenience we can always suppose that the block in Gi containing both wσ and wi is Bi1 if 1 ≤ i ≤ k (that is, Bi1 is the block of type I that belongs to both Gσ and Gi ), and is Bisi if k + 1 ≤ i ≤ θ . Thus, bearing in mind that wi ∈ NG (v ) and, by (3.12), (3.13) and (3.14), we have e(NG (v ), V (G)\NG (v )) = e(NGσ (v ), V (Gσ )\NGσ (v ))
+
k ( ti ∑ ∑ j=2
i=1
+
)
dBij (wi )
j=ti +1
ti θ (∑ ∑ i=k+1
si ∑
dBij (wi ) +
si −1 ∑
dBij (wi ) +
j=1
dBij (wi )
)
j=ti +1
⏐ ⏐ ≤ 2⏐V (Gσ )⏐ − 2(tσ + 1) − 2(sσ − tσ ) ) k ( ti ∑ ∑ + (nij − 2) + 2(si − ti ) j=2
i=1
+
θ ∑
(∑ ti
i=k+1
)
(nij − 2) + 2(si − ti − 1)
j=1
⏐ ⏐ ≤ n + ⏐V (Gσ )⏐ − 2(tσ + 1) − 2(sσ − tσ ) −
k ∑
θ ∑
i=1
i=k+1
(ti − 1) −
ti ,
(3.15)
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where the last inequality follows from the fact that
⏐
⏐
n ≥ ⏐V (Gσ )⏐ +
k θ ∑ ⏐ ⏐ ∑ ⏐ ⏐ ⏐V (Gi )\V (Bi1 )⏐ + ⏐V (Gi )\V (Bis )⏐ i
i=1
i=k+1
) k ( ti ∑ ⏐ ⏐ ∑ ⏐ ⏐ = V (Gσ ) + (nij − 1) + 2(si − ti ) i=1
+
j=2
ti θ (∑ ∑
)
(nij − 1) + 2(si − ti − 1) .
i=k+1
j=1
Now, by (3.10) and (3.15), we may conclude that 2e(NG (v )) + e(NG (v ), V (G)\NG (v )) k θ ∑ ∑ ⏐ ⏐ ≤ n + ⏐V (Gσ )⏐ − 2(tσ + 1) − (ti − 1) − ti i=1
i=k+1
⏐ ⏐ ≤ n + ⏐V (Gσ )⏐ − 2(tσ + 1),
(3.16)
where the last inequality follows from the fact that ti ≥ 0 for 1 ≤ i ≤ θ , and ti ≥ 1 for 1 ≤ i ≤ k (as Bi1 is always a block of type I in Gi ). Consequently, if tσ ≥ 1, then (3.2) follows directly from (3.16) and (3.8). If tσ = 0, then Gσ is a friendship ∗ graph with wσ being its center; furthermore, it follows from (3.8) that |V (Gσ )| < n − 2 as G ≇ F(n −1)/2 . This, together with (3.16), yields (3.2) again. Subcase 2.2. v is not any cut vertex of G. In this case, v only belongs to some block, say Bαβ , where α ∈ {1, . . . , σ } and β ∈ {1, 2, . . . , sα }. Clearly, NG (v ) = NBαβ (v ) and hence, e(NG (v )) = e(NBαβ (v )).
(3.17)
It is also easy to see that nαβ < n and 0 ≤ θ (v ) ≤ σ . If θ (v ) = 0, then noting that NG (u) ⊆ V (Bαβ ) for any vertex u ∈ NG (v ), we have e(NG (v ), V (G)\NG (v )) = e(NBαβ (v ), V (Bαβ )\NBαβ (v )).
(3.18)
Moreover, Bαβ must be type I and hence, from (3.3) and (3.4) it follows that e(NBαβ (v )) = 0 and e(NBαβ (v ), V (Bαβ )\NBαβ (v )) ≤ 2nαβ − 4 < 2n − 4. These, together with (3.17) and (3.18), would yield (3.2). If θ (v ) = θ ≥ 1, then without loss of generality, suppose that w1 , . . . , wθ are the cut vertices of G adjacent to v . Clearly, for each i ∈ {1, . . . , θ }, Bαβ is a block of Gi , and for convenience we sometimes regard Bαβ as the block Bi1 in Gi if Bαβ is type I, and as the block Bisi in Gi if Bαβ is type II. Thus, if Bαβ is type I, then there is at least one block of type I in Gi and hence, ti ≥ 1, i = 1, . . . , θ.
(3.19)
Moreover, we have
⏐
⏐
n ≥ ⏐V (Bαβ )⏐ +
θ ∑ ⏐ ⏐ ⏐V (Gi )\V (Bαβ )⏐ i=1
= nαβ +
ti θ (∑ ∑ i=1
)
(nij − 1) + 2(si − ti ) .
(3.20)
j=2
Consequently, by (3.17) and (3.3), we obtain e(NG (v )) = e(NBαβ (v )) = 0, and by (3.4), (3.13), (3.14), (3.20) and (3.19), we get e(NG (v ), V (G)\NG (v )) = e(NBαβ (v ), V (Bαβ )\NBαβ (v ))
+
ti θ (∑ ∑ i=1
dBij (wi ) +
j=2
≤ 2nαβ − 4 +
si ∑
dBij (wi )
)
j=ti +1 ti θ (∑ ∑ i=1
j=2
) (nij − 2) + 2(si − ti )
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X. Chen and L. Guo / Discrete Mathematics 342 (2019) 2092–2099
≤ n + nαβ − 4 −
θ ∑
(ti − 1)
i=1
≤ n + nαβ − 4, which yields (3.2), as nαβ < n. If Bαβ is type II, then
⏐
θ ∑ ⏐ ⏐ ⏐V (Gi )\V (Bαβ )⏐
⏐
n ≥ ⏐V (Bαβ )⏐ +
i=1
= 3+
ti θ (∑ ∑ i=1
)
(nij − 1) + 2(si − ti − 1) .
(3.21)
j=1
Consequently, by (3.17) and (3.5), we have e(NG (v )) = e(NBαβ (v )) = 1, and by (3.6), (3.13), (3.14), and (3.21), we get e(NG (v ), V (G)\NG (v )) = e(NBαβ (v ), V (Bαβ )\NBαβ (v ))
+
ti θ (∑ ∑ i=1
≤ 2+
dBij (wi ) +
≤ n−1−
)
dBij (wi )
j=ti +1
j=1
ti θ (∑ ∑ i=1
si −1 ∑
) (nij − 2) + 2(si − ti − 1)
j=1
θ ∑
ti ,
i=1
which yields (3.2) again, since ti ≥ 0 for i = 1, . . . , θ , and n > 5 (as G ≇ F2 ). The proof is completed. □ We are now ready to present the proof of Theorem 1.3. Proof of Theorem 1.3. Let G be a minimally 2-edge-connected graph of order n ≥ 4. Clearly, by Lemmas 2.1 and 3.1, we obtain ρ (G) ≥ δ (G) = 2, with equality if and only if G is 2-regular, that is, G ∼ = Cn . λ ∗ We next turn to the upper bound on ρ (G). If G ≇ F(n−1)/2 , K2, n−2 , F(n −1)/2 and Fp, (n−p−2)/2 (λ = 1, 2), then by (2.1) as
√
well as Lemmas 2.2 and 3.2, we have ρ (G) < 2n − 4. On the other hand, noting that n ≥ 7 and p < n − 2 (required in λ ∗ the definitions of the graphs F(n −1)/2 and Fp, (n−p−2)/2 (λ = 1, 2), respectively), and using Lemma 2.2, we can easily deduce that
ρ (F(n∗ −1)/2 ) < max ρ
(Fp1, (n−p−2)/2 )
ρ
(Fp2, (n−p−2)/2 )
{√
2n − 4,
< max < max
{√
{√
√
n + 3,
2n − 4,
√
n + 1,
√
√
√
√
√
n − 2 + p,
n − 1,
n,
√ }
√ }
6 =
√
√
2n − 4,
2n − 4, √ } √ 2n − 4, n − 2 + p, n − p + 2, 2p = 2n − 4. 2p =
It is also known that
ρ (K2, n−2 ) = ρ (F(n−1)/2 ) =
√
2n − 4, (see [9], p. 72)
√
1+
4n − 3 2
. (see [5])
Further, a simple calculation shows that ρ (K2, n−2 ) > ρ (F(n−1)/2 ) when n ≥ 6, and ρ (K √2, n−2 ) < ρ (F(n−1)/2 ) when n = 5. Now, from the above arguments, we √ may conclude that if n ̸ = 5, then ρ (G) ≤ 2n − 4 with equality if and only if G∼ = K2, n−2 , and if n = 5, then ρ (G) ≤ 1+2 17 with equality if and only if G ∼ = F2 , as desired. This completes the proof of Theorem 1.3. □ 4. Concluding remark In this paper, we present a complete characterization of the graphs with extremal spectral radius among all minimally 2-(edge)-connected graphs. It would be of interest to further characterize the graphs with extremal spectral radius among
X. Chen and L. Guo / Discrete Mathematics 342 (2019) 2092–2099
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all minimally k-(edge)-connected graphs for k ≥ 3. We notice that there is direct lower bound on ρ (G) for a minimally k-(edge)-connected graph G, that is, ρ (G) ≥ k, which follows immediately from Lemma 2.1 and the fact that if G is minimally k-(edge)-connected, then δ (G) = k (see [28], p. 175). However, when k ≥ 3, it seems not easy to completely characterize the graphs achieving this lower bound. We would leave this problem for future research. On the other hand, a complete characterization of the minimally k-(edge)-connected graphs with maximal spectral radius for k ≥ 3 seems more difficult. We end this paper by mentioning another application of Lemmas 2.6 and 3.2. Let M(G) denote the sum of squares of degrees of a graph G. It is easy to see that M(G) =
∑ v∈V (G)
dG (v )2 =
∑
∑
dG (u),
v∈V (G) u∈NG (v )
from which, as well as (2.1) and Lemmas 2.4, 2.6 and 3.2, we may easily conclude the following two results. Theorem 4.1. G∼ = K2, n−2 .
If G is a minimally 2-connected graph of order n ≥ 4, then M(G) ≤ 2n(n − 2), with equality if and only if
Theorem 4.2. Let G be a minimally 2-edge-connected graph of order n ≥ 4. If n ̸ = 5, then M(G) ≤ 2n(n − 2), with equality if and only if G ∼ = K2, n−2 ; if n = 5, then M(G) ≤ (n − 1)(n + 3), with equality if and only if G ∼ = F2 . Acknowledgments The authors would like to thank the anonymous referees for their valuable comments and suggestions which have contributed to the final preparation of the paper. This work was supported by Natural Science Foundation of Fujain Province, China (NO. 2019J01857), National Natural Science Foundation of China (Nos. 11861011, 11501133) and Natural Science Foundation of Guangxi Province, China (No. 2016GXNSFAA380293). Conflict of interest statement None. Declaration of conflicting of interests The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30]
L. Babai, B. Guiduli, Spectral extrema for graphs: the Zarankiewicz problem, Electron. J. Combin. 16 (2009) #R123. A. Berman, X. Zhang, On the spectral radius of graphs with cut veritces, J. Combin. Theory Ser. B 83 (2001) 233–240. J.A. Bondy, U.S.R. Murty, Graph Theory, in: GTM, vol. 244, Springer, New York (NY), 2008. R.A. Brualdi, E.S. Solheid, On the spectral radius of complementary acyclic matrices of zeros and ones, SIAM J. Algebra. Discrete Method 7 (1986) 265–272. K. Ch. Das, Proof of conjectures on adjacency eigenvalues of graphs, Discrete Math. 313 (2013) 19–25. A. Chang, F. Tian, On the spectral radius of unicyclic graphs with perfect matchings, Linear Algebra Appl. 370 (2003) 237–250. A. Chang, F. Tian, On the index of bicyclic graphs with perfect matchings, Discrete Math. 283 (2004) 51–59. L. Collatz, U. Sinogowitz, Spektren endlicher graphen, Abh. Math. Semin. Univ. Hambg. 21 (1957) 63–77 (in German). D. Cvetković, M. Doob, H. Sachs, Spectra of Graphs, Academic Press, New York, 1980. D. Cvetković, P. Rowlinson, The largest eigenvalue of a graph: a survey, Linear Multilinear Algebra 28 (1990) 3–33. E.R. van Dam, Graphs with given diameter maximizing the spectral radius, Linear Algebra Appl. 426 (2007) 454–457. E.R. van Dam, R.E. Kooij, The minimal spectral radius of graphs with a given diameter, Linear Algebra Appl. 423 (2007) 408–419. A. Dirac, Minimally 2-connected graphs, J. Reine Angew. Math. 228 (1967) 204–216. O. Favaron, M. Mahéo, J.-F. Saclé, Some eigenvalue properties in graphs (conjectures of Graffiti, II), Discrete Math. 111 (1993) 197–220. L. Feng, Q. Li, X. Zhang, Spectral radii of graphs with given chromatic number, Appl. Math. Lett. 20 (2007) 158–162. L. Feng, G. Yu, X. Zhang, Spectral radius of graphs with given matching number, Linear Algebra Appl. 422 (2007) 133–138. J. Guo, J. Shao, On the spectral radius of trees with fixed diameter, Linear Algebra Appl. 413 (2006) 131–147. P. Hansen, D. Stevanović, On bags and bugs, Discrete Appl. Math. 156 (2008) 986–997. H. Liu, M. Lu, On the spectral radius of quasi-tree graphs, Linear Algebra Appl. 428 (2008) 2708–2714. H. Liu, M. Lu, F. Tian, On the spectral radius of graphs with cut edges, Linear Algebra Appl. 389 (2004) 139–145. V. Nikiforov, Bounds on graph eigenvalues II, Linear Algebra Appl. 427 (2007) 183–189. V. Nikiforov, The maximum spectral radius of C4 -free graphs of given order and size, Linear Algebra Appl. 430 (2009) 2898–2905. V. Nikiforov, The spectral radius of graphs without paths and cycles of specified length, Linear Algebra Appl. 432 (2010) 2243–2256. D. Stevanović, Spectral Radius of Graphs, Academic Press, Amsterdam, 2015. D. Stevanović, M. Aouchiche, P. Hansen, On the spectral radius of graphs with a given domination number, Linear Algebra Appl. 428 (2008) 1854–1864. D. Stevanović, P. Hansen, The minimum spectral radius of graphs with a given clique number, Electron. J. Linear Algebra 17 (2008) 110–117. M. Tait, J. Tobin, Three conjectures in extremal spectral graph theory, J. Combin. Theory Ser. B 126 (2017) 137–161. D.B. West, Introduction To Graph Theory, second ed., Prentice-Hall, Upper Saddle River (NJ), 2001. B. Wu, E. Xiao, Y. Hong, The spectral radius of trees on k pendant vertices, Linear Algebra Appl. 395 (2005) 343–349. M. Xu, Y. Hong, J. Shu, M. Zhai, The minimum spectral radius of graphs with a given independence number, Linear Algebra Appl. 431 (2009) 937–945.
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Discrete Mathematics journal homepage: www.elsevier.com/locate/disc
On minimally 2-(edge)-connected graphs with extremal spectral radius ∗
Xiaodan Chen a , Litao Guo b , a b
College of Mathematics and Information Science, Guangxi University, Nanning 530004, Guangxi, PR China School of Applied Mathematics, Xiamen University of Technology, Xiamen 361024, Fujian, PR China
article
info
Article history: Received 7 September 2018 Received in revised form 24 January 2019 Accepted 1 April 2019 Available online 30 April 2019
a b s t r a c t A graph is minimally 2-(edge)-connected if it is 2-(edge)-connected and deleting any arbitrary chosen edge always leaves a graph which is not 2-(edge)-connected. In this paper, we completely characterize the minimally 2-(edge)-connected graphs having the largest and the smallest spectral radius, respectively. © 2019 Elsevier B.V. All rights reserved.
Keywords: Minimally 2-connected graph Minimally 2-edge-connected graph Spectral radius Extremal graph
1. Introduction Throughout this paper we consider finite, undirected and simple graphs. Let G be a graph with vertex set V (G) = {v1 , v2 , . . . , vn } and edge set E(G). For i ∈ {1, 2, . . . , n}, denote by NG (vi ) and dG (vi ) the neighborhood and the degree of the vertex vi in G, respectively. A graph G is said to be r-regular if dG (vi ) = r for each i ∈ {1, 2, . . . , n}. We also write ∆(G) and δ (G) for the maximum degree and the minimum degree of G, respectively. As usual, let Kn , Cn and Pn denote the complete graph, the cycle and the path on n vertices, respectively. Let Ka, b , also, denote the complete bipartite graph with two partite sets having a and b vertices, respectively. The adjacency matrix of a graph G is defined as the n × n square matrix A(G) = (aij ) whose entries are given by
{ aij =
1 0
if vi vj ∈ E(G), otherwise.
The spectral radius of G, denoted by ρ (G), is defined to be the spectral radius of A(G), i.e., the largest eigenvalue of A(G). The investigation on the spectral radius of graphs, which could date back to the seminal work of Collatz and Sinogowitz [8] in the 1950s, has been a central research theme in spectral graph theory. There have been numerous results concerning the spectral radius of graphs; for details one may refer to a survey by Cvetković and Rowlinson [10] in 1990 and a recent research monograph by Stevanović [24]. It is noted that most of these results relate to the famous Brualdi– Solheid’s general question [4] that asks to characterize graphs with extremal values of the spectral radius in a given class of graphs (where ‘‘extremal’’ usually means ‘‘maximal’’); see [1,2,6,7,11,12,15–23,25–27,29,30] for more information. We here mention a classical result of this type. ∗ Corresponding author. E-mail addresses: [email protected] (X. Chen), [email protected] (L. Guo). https://doi.org/10.1016/j.disc.2019.04.002 0012-365X/© 2019 Elsevier B.V. All rights reserved.
X. Chen and L. Guo / Discrete Mathematics 342 (2019) 2092–2099
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Fig. 1. The graphs Fq , Fq∗ , Fp1, q and Fp2, q .
Theorem 1.1 (See [14]). If T is a tree of order n ≥ 2, then 2 cos
π n+1
≤ ρ (T ) ≤
√
n − 1,
with equality in the left (resp., right) inequality if and only if T ∼ = Pn (resp., T ∼ = K1, n−1 ). A graph is k-connected if removing fewer than k vertices always leaves the remaining graph connected, and is minimally k-connected if it is k-connected and deleting any arbitrary chosen edge always leaves a graph which is not k-connected. Analogously, a graph is k-edge-connected if removing fewer than k edges always leaves the remaining graph connected, and is minimally k-edge-connected if it is k-edge-connected and deleting any arbitrary chosen edge always leaves a graph which is not k-edge-connected. Observe that a graph is minimally 1-(edge)-connected if and only if it is a tree. So, it is natural to ask which graphs have the extremal spectral radius among all minimally k-(edge)-connected graphs for k ≥ 2. In this paper, we are concerned with the extremal spectral radius of minimally 2-(edge)-connected graphs and obtain the following two results. Theorem 1.2.
If G is a minimally 2-connected graph of order n ≥ 4, then
2 ≤ ρ (G) ≤
√
2n − 4,
with equality in the left (resp., right) inequality if and only if G ∼ = Cn (resp., G ∼ = K2, n−2 ). Theorem 1.3. Let G be a minimally 2-edge-connected graph of order n ≥ 4. (i) ρ (G) ≥ 2, with equality if and only if G ∼ = Cn . √ (ii) If n ̸ = 5, then ρ (G) ≤ 2n − 4, with equality if and only if G ∼ = K2, n−2 ; if n = 5, then ρ (G) ≤ and only if G ∼ = F2 , where F2 is the friendship graph having two triangles (see Fig. 1).
√ 1+ 17 , 2
with equality if
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The proofs of Theorems 1.2 and 1.3 will be presented in the forthcoming two sections, and a brief remark will be given in the final section. 2. Proof of Theorem 1.2 In order to give the proof of Theorem 1.2, we need some auxiliary results. The first one is well known, which can be found in [9], pp. 84–85. Lemma 2.1 (See [9]). For any graph G, we have ρ (G) ≥ δ (G), with equality if and only if G is regular. The next result comes from Favaron, Mahéo and Saclé [14]. Lemma 2.2 (See [14]). If G is a connected graph, then
ρ (G) ≤ max
v∈V (G)
√ ∑
dG (u).
u∈NG (v )
Equality holds if and only if G is regular or bipartite semiregular. For X , Y ⊂ V (G), we denote by e(X ) the number of edges in the subgraph induced by X and by e(X , Y ) the number of edges with one endpoint in X and one endpoint in Y . It is easy to check that (see also [27])
∑
dG (u) = 2e(NG (v )) + e(NG (v ), V (G)\NG (v )).
(2.1)
u∈NG (v )
We also need some classical results concerning the structure of minimally 2-connected graphs. Lemma 2.3 (See [3], p. 211). If G is a minimally 2-connected graph, then δ (G) = 2. Lemma 2.4 (See [13]). A minimally 2-connected graph with more than three vertices contains no triangles. Lemma 2.5 (See [13]). A minimally 2-connected graph G with n ≥ 4 vertices has as most 2n − 4 edges, with equality if and only if G ∼ = K2, n−2 . The following result is a direct consequence of Lemma 2.5. Lemma 2.6.
Let G be a minimally 2-connected graph of order n ≥ 4. Then for any vertex v ∈ V (G),
e(NG (v ), V (G)\NG (v )) ≤ 2n − 4, with equality if and only if G ∼ = K2, n−2 . We are now ready to give the proof of Theorem 1.2. Proof of Theorem 1.2. Let G be a minimally 2-connected graph of order n ≥ 4. Clearly, by Lemmas 2.1 and 2.3, we have ρ (G) ≥ δ (G) = 2, with equality if and only if G is 2-regular, that is, G ∼ =√Cn . 2n − 4 when G ≇ K2, n−2 , because ρ (K2, n−2 ) = For the upper bound on ρ (G), we just need to show that ρ (G) < √ 2n − 4 (see [9], p. 72). Suppose now that G ≇ K2, n−2 . For any vertex v ∈ V (G), by Lemma 2.6, we obtain e(NG (v ), V (G)\NG (v )) <√2n − 4. Moreover, by Lemma 2.4, we have e(NG (v )) = 0. Consequently, from (2.1) and Lemma 2.2 it follows that ρ (G) < 2n − 4, as desired. This completes the proof of Theorem 1.2. □ 3. Proof of Theorem 1.3 For proving Theorem 1.3, we need two necessary results. Lemma 3.1 (See [3], p. 219). If G is a minimally 2-edge-connected graph, then δ (G) = 2. For q ≥ 1, let Fq be the friendship graph of order n = 2q + 1, that is, the graph consisting of q triangle(s) intersecting in a common vertex. Let Fq∗ be the graph of order n = 2q + 1 (q ≥ 3) obtained from Fq−1 by coalescing one of its noncentral vertices and a vertex of K3 . Also, for p ≥ 2 and q ≥ 1, let Fp1, q (resp., Fp2, q ) be the graph of order n = p + 2q + 2 obtained from Fq by coalescing its center x and a p-degree (resp., 2-degree) vertex of K2, p . Fig. 1 shows the graphs Fq , Fq∗ , Fp1, q and Fp2, q ; obviously, all these graphs are minimally 2-edge-connected.
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Lemma 3.2.
2095
Let G be a minimally 2-edge-connected graph of order n ≥ 4. If G ≇ F(n−1)/2 , then for any vertex v ∈ V (G),
2e(NG (v )) + e(NG (v ), V (G)\NG (v )) ≤ 2n − 4.
(3.1) Fpλ, (n−p−2)/2
Moreover, the equality holds in (3.1) if and only if G ∼ (λ = 1, 2) and v = K2, n−2 (and v is any vertex of G), or G ∼ = is the unique cut vertex of G, or G ∼ = F(n∗ −1)/2 and v is the cut vertex of G whose degree is as large as possible. Proof. If G ∼ = K2, n−2 , then it can be directly checked that the equality holds in (3.1) for any vertex v ∈ V (G). It is also not difficult to verify that if G ∼ = Fpλ, (n−p−2)/2 (λ = 1, 2), then the equality holds in (3.1) only for the unique cut vertex of G, ∗ ∼ and that if G = F(n−1)/2 , then the equality holds in (3.1) only for the cut vertex of G whose degree is as large as possible. Therefore, to complete the proof, we just need to show that 2e(NG (v )) + e(NG (v ), V (G)\NG (v )) < 2n − 4
(3.2)
∗ holds for any vertex v ∈ V (G) when G ≇ F(n−1)/2 , K2, n−2 , Fpλ, (n−p−2)/2 (λ = 1, 2) and F(n −1)/2 . ∗ . We denote by σ the number of cut vertices Suppose now that G ≇ F(n−1)/2 , K2, n−2 , Fpλ, (n−p−2)/2 (λ = 1, 2) and F(n −1)/2 in G and consider the following two cases. Case 1. σ = 0. In this case, G is 2-connected. Furthermore, G is minimally 2-connected, because G is minimally 2-edge-connected. Thus, for any vertex v ∈ V (G), by Lemma 2.4, we have e(NG (v )) = 0. This, together with Lemma 2.6 and the assumption that G ≇ K2, n−2 , yields (3.2) immediately. Case 2. σ ≥ 1. Let w1 , . . . , wσ denote the cut vertices of G. For each i ∈ {1, . . . , σ }, let Bi1 , Bi2 , . . . , Bisi be all the blocks of G which contain wi ; note that si ≥ 2. Clearly, for each j ∈ {1, 2, . . . , si }, Bij is minimally 2-edge-connected, since G is so. Furthermore, because G is simple, we have |V (Bij )| = nij ≥ 3 and then, Bij is minimally 2-connected. We will say that a block Bij is type I if nij ≥ 4, and is type II if nij = 3. For any vertex v ∈ V (Bij ), if Bij is type I, then by Lemmas 2.4 and 2.6, we obtain
e(NBij (v )) = 0,
(3.3)
e(NBij (v ), V (Bij )\NBij (v )) ≤ 2nij − 4,
(3.4)
and
with equality if and only if Bij ∼ = K3 and consequently, we have = K2, nij −2 ; if Bij is type II, then it is easy to see that Bij ∼ e(NBij (v )) = 1,
(3.5)
e(NBij (v ), V (Bij )\NBij (v )) = 2.
(3.6)
and
Set Gi := Bi1 ∪ Bi2 ∪ · · · ∪ Bisi . Let ti be the number of blocks of type I in Gi and without loss of generality, suppose that Bi1 , Bi2 , . . . , Biti are such blocks. Clearly, 0 ≤ ti ≤ si and there are no blocks of type I in Gi if ti = 0. It is also easy to see that Gi is a (not necessary proper) subgraph of G with wi being its unique cut vertex and, ti ⏐ ⏐ ∑ ⏐V (Gi )⏐ = nij − (ti − 1) + 2(si − ti ).
(3.7)
j=1
Now, for any vertex v ∈ V (G), we denote by θ (v ) the number of the cut vertices of G adjacent to v , and divide into the following two subcases. Subcase 2.1. v is one of the cut vertices of G. Without loss of generality, we can suppose that v = wσ . Clearly, if σ = 1, then Gσ ∼ = G and hence, |V (Gσ )| = n; if σ ≥ 2, then
⏐ ⏐ ⏐V (Gσ )⏐ ≤ n − 2,
(3.8)
with equality if and only if G is the graph obtained by coalescing any non-cut vertex of Gσ and a vertex of K3 (in this case,
σ = 2). It is also observed that (⋃ ) ⋃( ⋃ ) tσ sσ NG (v ) = NGσ (v ) = NBσ j (v ) NBσ j (v ) , j=1
(3.9)
j=tσ +1
and NBσ i (v ) NBσ j (v ) = ∅ for any 1 ≤ i ̸ = j ≤ sσ . Since each edge of G only belongs to one block of G, by (3.9), (3.3) and (3.5), we have
⋂
e(NG (v )) = e(NGσ (v )) =
tσ ∑ j=1
e(NBσ j (v )) +
sσ ∑ j=tσ +1
e(NBσ j (v )) = sσ − tσ .
(3.10)
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Note that 0 ≤ θ (v ) ≤ σ − 1. If θ (v ) = 0, then tσ ≥ 1 (as G ≇ F(n−1)/2 and any cut vertex in a block of type II in Gσ is always adjacent to v ). Moreover, observing that NG (u) ⊆ V (Gσ ) for any vertex u ∈ NG (v ), we obtain e(NG (v ), V (G)\NG (v )) = e(NGσ (v ), V (Gσ )\NGσ (v )).
(3.11)
We further observe that NGσ (u) ⊆ V (Bσ j ) for any vertex u ∈ NBσ j (v ), which, together with (3.9), (3.4), (3.6) and (3.7), yields that e(NGσ (v ), V (Gσ )\NGσ (v ))
=
tσ ∑
e(NBσ j (v ), V (Bσ j )\NBσ j (v ))
j=tσ +1
j=1
≤
sσ ∑
e(NBσ j (v ), V (Bσ j )\NBσ j (v )) +
tσ ∑
(2nσ j − 4) + 2(sσ − tσ )
j=1
⏐ ⏐ = 2⏐V (Gσ )⏐ − 2(tσ + 1) − 2(sσ − tσ ),
(3.12)
with equality if and only if Bσ j ∼ = K2, nσ j −2 for all j ∈ {1, . . . , tσ }. We here remark that if σ = 1 and tσ = 1, then the strict inequality holds in (3.12), because G ≇ Fpλ, (n−p−2)/2 (λ = 1, 2). Now, from (3.10), (3.11) and (3.12), it follows that 2e(NG (v )) + e(NG (v ), V (G)\NG (v )) ≤ 2⏐V (Gσ )⏐ − 2(tσ + 1),
⏐
⏐
with the strict inequality if σ = 1 and tσ = 1. This, together with the fact that tσ ≥ 1 and |V (Gσ )| ≤ n with equality if and only if σ = 1, yields (3.2). If θ (v ) = θ ≥ 1, then σ ≥ 2. Moreover, we suppose, without loss of generality, that w1 , . . . , wk belong to some blocks of type I in Gσ , and wk+1 , . . . , wθ belong to some blocks of type II in Gσ ; note that 0 ≤ k ≤ θ and, w1 , . . . , wθ all belong to some blocks of type I in Gσ if k = θ , and all belong to some blocks of type II in Gσ if k = 0. Next, for each i ∈ {1, . . . , θ }, consider Gi and its blocks Bi1 , Bi2 , . . . , Bisi . Recalling that Bij is minimally-2-connected, if Bij is type I (in this case, 1 ≤ j ≤ ti ), then dBij (wi ) ≤ nij − 2,
(3.13)
and if Bij is type II (in this case, ti + 1 ≤ j ≤ si ), then dBij (wi ) = 2.
(3.14)
Moreover, for convenience we can always suppose that the block in Gi containing both wσ and wi is Bi1 if 1 ≤ i ≤ k (that is, Bi1 is the block of type I that belongs to both Gσ and Gi ), and is Bisi if k + 1 ≤ i ≤ θ . Thus, bearing in mind that wi ∈ NG (v ) and, by (3.12), (3.13) and (3.14), we have e(NG (v ), V (G)\NG (v )) = e(NGσ (v ), V (Gσ )\NGσ (v ))
+
k ( ti ∑ ∑ j=2
i=1
+
)
dBij (wi )
j=ti +1
ti θ (∑ ∑ i=k+1
si ∑
dBij (wi ) +
si −1 ∑
dBij (wi ) +
j=1
dBij (wi )
)
j=ti +1
⏐ ⏐ ≤ 2⏐V (Gσ )⏐ − 2(tσ + 1) − 2(sσ − tσ ) ) k ( ti ∑ ∑ + (nij − 2) + 2(si − ti ) j=2
i=1
+
θ ∑
(∑ ti
i=k+1
)
(nij − 2) + 2(si − ti − 1)
j=1
⏐ ⏐ ≤ n + ⏐V (Gσ )⏐ − 2(tσ + 1) − 2(sσ − tσ ) −
k ∑
θ ∑
i=1
i=k+1
(ti − 1) −
ti ,
(3.15)
X. Chen and L. Guo / Discrete Mathematics 342 (2019) 2092–2099
2097
where the last inequality follows from the fact that
⏐
⏐
n ≥ ⏐V (Gσ )⏐ +
k θ ∑ ⏐ ⏐ ∑ ⏐ ⏐ ⏐V (Gi )\V (Bi1 )⏐ + ⏐V (Gi )\V (Bis )⏐ i
i=1
i=k+1
) k ( ti ∑ ⏐ ⏐ ∑ ⏐ ⏐ = V (Gσ ) + (nij − 1) + 2(si − ti ) i=1
+
j=2
ti θ (∑ ∑
)
(nij − 1) + 2(si − ti − 1) .
i=k+1
j=1
Now, by (3.10) and (3.15), we may conclude that 2e(NG (v )) + e(NG (v ), V (G)\NG (v )) k θ ∑ ∑ ⏐ ⏐ ≤ n + ⏐V (Gσ )⏐ − 2(tσ + 1) − (ti − 1) − ti i=1
i=k+1
⏐ ⏐ ≤ n + ⏐V (Gσ )⏐ − 2(tσ + 1),
(3.16)
where the last inequality follows from the fact that ti ≥ 0 for 1 ≤ i ≤ θ , and ti ≥ 1 for 1 ≤ i ≤ k (as Bi1 is always a block of type I in Gi ). Consequently, if tσ ≥ 1, then (3.2) follows directly from (3.16) and (3.8). If tσ = 0, then Gσ is a friendship ∗ graph with wσ being its center; furthermore, it follows from (3.8) that |V (Gσ )| < n − 2 as G ≇ F(n −1)/2 . This, together with (3.16), yields (3.2) again. Subcase 2.2. v is not any cut vertex of G. In this case, v only belongs to some block, say Bαβ , where α ∈ {1, . . . , σ } and β ∈ {1, 2, . . . , sα }. Clearly, NG (v ) = NBαβ (v ) and hence, e(NG (v )) = e(NBαβ (v )).
(3.17)
It is also easy to see that nαβ < n and 0 ≤ θ (v ) ≤ σ . If θ (v ) = 0, then noting that NG (u) ⊆ V (Bαβ ) for any vertex u ∈ NG (v ), we have e(NG (v ), V (G)\NG (v )) = e(NBαβ (v ), V (Bαβ )\NBαβ (v )).
(3.18)
Moreover, Bαβ must be type I and hence, from (3.3) and (3.4) it follows that e(NBαβ (v )) = 0 and e(NBαβ (v ), V (Bαβ )\NBαβ (v )) ≤ 2nαβ − 4 < 2n − 4. These, together with (3.17) and (3.18), would yield (3.2). If θ (v ) = θ ≥ 1, then without loss of generality, suppose that w1 , . . . , wθ are the cut vertices of G adjacent to v . Clearly, for each i ∈ {1, . . . , θ }, Bαβ is a block of Gi , and for convenience we sometimes regard Bαβ as the block Bi1 in Gi if Bαβ is type I, and as the block Bisi in Gi if Bαβ is type II. Thus, if Bαβ is type I, then there is at least one block of type I in Gi and hence, ti ≥ 1, i = 1, . . . , θ.
(3.19)
Moreover, we have
⏐
⏐
n ≥ ⏐V (Bαβ )⏐ +
θ ∑ ⏐ ⏐ ⏐V (Gi )\V (Bαβ )⏐ i=1
= nαβ +
ti θ (∑ ∑ i=1
)
(nij − 1) + 2(si − ti ) .
(3.20)
j=2
Consequently, by (3.17) and (3.3), we obtain e(NG (v )) = e(NBαβ (v )) = 0, and by (3.4), (3.13), (3.14), (3.20) and (3.19), we get e(NG (v ), V (G)\NG (v )) = e(NBαβ (v ), V (Bαβ )\NBαβ (v ))
+
ti θ (∑ ∑ i=1
dBij (wi ) +
j=2
≤ 2nαβ − 4 +
si ∑
dBij (wi )
)
j=ti +1 ti θ (∑ ∑ i=1
j=2
) (nij − 2) + 2(si − ti )
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X. Chen and L. Guo / Discrete Mathematics 342 (2019) 2092–2099
≤ n + nαβ − 4 −
θ ∑
(ti − 1)
i=1
≤ n + nαβ − 4, which yields (3.2), as nαβ < n. If Bαβ is type II, then
⏐
θ ∑ ⏐ ⏐ ⏐V (Gi )\V (Bαβ )⏐
⏐
n ≥ ⏐V (Bαβ )⏐ +
i=1
= 3+
ti θ (∑ ∑ i=1
)
(nij − 1) + 2(si − ti − 1) .
(3.21)
j=1
Consequently, by (3.17) and (3.5), we have e(NG (v )) = e(NBαβ (v )) = 1, and by (3.6), (3.13), (3.14), and (3.21), we get e(NG (v ), V (G)\NG (v )) = e(NBαβ (v ), V (Bαβ )\NBαβ (v ))
+
ti θ (∑ ∑ i=1
≤ 2+
dBij (wi ) +
≤ n−1−
)
dBij (wi )
j=ti +1
j=1
ti θ (∑ ∑ i=1
si −1 ∑
) (nij − 2) + 2(si − ti − 1)
j=1
θ ∑
ti ,
i=1
which yields (3.2) again, since ti ≥ 0 for i = 1, . . . , θ , and n > 5 (as G ≇ F2 ). The proof is completed. □ We are now ready to present the proof of Theorem 1.3. Proof of Theorem 1.3. Let G be a minimally 2-edge-connected graph of order n ≥ 4. Clearly, by Lemmas 2.1 and 3.1, we obtain ρ (G) ≥ δ (G) = 2, with equality if and only if G is 2-regular, that is, G ∼ = Cn . λ ∗ We next turn to the upper bound on ρ (G). If G ≇ F(n−1)/2 , K2, n−2 , F(n −1)/2 and Fp, (n−p−2)/2 (λ = 1, 2), then by (2.1) as
√
well as Lemmas 2.2 and 3.2, we have ρ (G) < 2n − 4. On the other hand, noting that n ≥ 7 and p < n − 2 (required in λ ∗ the definitions of the graphs F(n −1)/2 and Fp, (n−p−2)/2 (λ = 1, 2), respectively), and using Lemma 2.2, we can easily deduce that
ρ (F(n∗ −1)/2 ) < max ρ
(Fp1, (n−p−2)/2 )
ρ
(Fp2, (n−p−2)/2 )
{√
2n − 4,
< max < max
{√
{√
√
n + 3,
2n − 4,
√
n + 1,
√
√
√
√
√
n − 2 + p,
n − 1,
n,
√ }
√ }
6 =
√
√
2n − 4,
2n − 4, √ } √ 2n − 4, n − 2 + p, n − p + 2, 2p = 2n − 4. 2p =
It is also known that
ρ (K2, n−2 ) = ρ (F(n−1)/2 ) =
√
2n − 4, (see [9], p. 72)
√
1+
4n − 3 2
. (see [5])
Further, a simple calculation shows that ρ (K2, n−2 ) > ρ (F(n−1)/2 ) when n ≥ 6, and ρ (K √2, n−2 ) < ρ (F(n−1)/2 ) when n = 5. Now, from the above arguments, we √ may conclude that if n ̸ = 5, then ρ (G) ≤ 2n − 4 with equality if and only if G∼ = K2, n−2 , and if n = 5, then ρ (G) ≤ 1+2 17 with equality if and only if G ∼ = F2 , as desired. This completes the proof of Theorem 1.3. □ 4. Concluding remark In this paper, we present a complete characterization of the graphs with extremal spectral radius among all minimally 2-(edge)-connected graphs. It would be of interest to further characterize the graphs with extremal spectral radius among
X. Chen and L. Guo / Discrete Mathematics 342 (2019) 2092–2099
2099
all minimally k-(edge)-connected graphs for k ≥ 3. We notice that there is direct lower bound on ρ (G) for a minimally k-(edge)-connected graph G, that is, ρ (G) ≥ k, which follows immediately from Lemma 2.1 and the fact that if G is minimally k-(edge)-connected, then δ (G) = k (see [28], p. 175). However, when k ≥ 3, it seems not easy to completely characterize the graphs achieving this lower bound. We would leave this problem for future research. On the other hand, a complete characterization of the minimally k-(edge)-connected graphs with maximal spectral radius for k ≥ 3 seems more difficult. We end this paper by mentioning another application of Lemmas 2.6 and 3.2. Let M(G) denote the sum of squares of degrees of a graph G. It is easy to see that M(G) =
∑ v∈V (G)
dG (v )2 =
∑
∑
dG (u),
v∈V (G) u∈NG (v )
from which, as well as (2.1) and Lemmas 2.4, 2.6 and 3.2, we may easily conclude the following two results. Theorem 4.1. G∼ = K2, n−2 .
If G is a minimally 2-connected graph of order n ≥ 4, then M(G) ≤ 2n(n − 2), with equality if and only if
Theorem 4.2. Let G be a minimally 2-edge-connected graph of order n ≥ 4. If n ̸ = 5, then M(G) ≤ 2n(n − 2), with equality if and only if G ∼ = K2, n−2 ; if n = 5, then M(G) ≤ (n − 1)(n + 3), with equality if and only if G ∼ = F2 . Acknowledgments The authors would like to thank the anonymous referees for their valuable comments and suggestions which have contributed to the final preparation of the paper. This work was supported by Natural Science Foundation of Fujain Province, China (NO. 2019J01857), National Natural Science Foundation of China (Nos. 11861011, 11501133) and Natural Science Foundation of Guangxi Province, China (No. 2016GXNSFAA380293). Conflict of interest statement None. Declaration of conflicting of interests The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30]
L. Babai, B. Guiduli, Spectral extrema for graphs: the Zarankiewicz problem, Electron. J. Combin. 16 (2009) #R123. A. Berman, X. Zhang, On the spectral radius of graphs with cut veritces, J. Combin. Theory Ser. B 83 (2001) 233–240. J.A. Bondy, U.S.R. Murty, Graph Theory, in: GTM, vol. 244, Springer, New York (NY), 2008. R.A. Brualdi, E.S. Solheid, On the spectral radius of complementary acyclic matrices of zeros and ones, SIAM J. Algebra. Discrete Method 7 (1986) 265–272. K. Ch. Das, Proof of conjectures on adjacency eigenvalues of graphs, Discrete Math. 313 (2013) 19–25. A. Chang, F. Tian, On the spectral radius of unicyclic graphs with perfect matchings, Linear Algebra Appl. 370 (2003) 237–250. A. Chang, F. Tian, On the index of bicyclic graphs with perfect matchings, Discrete Math. 283 (2004) 51–59. L. Collatz, U. Sinogowitz, Spektren endlicher graphen, Abh. Math. Semin. Univ. Hambg. 21 (1957) 63–77 (in German). D. Cvetković, M. Doob, H. Sachs, Spectra of Graphs, Academic Press, New York, 1980. D. Cvetković, P. Rowlinson, The largest eigenvalue of a graph: a survey, Linear Multilinear Algebra 28 (1990) 3–33. E.R. van Dam, Graphs with given diameter maximizing the spectral radius, Linear Algebra Appl. 426 (2007) 454–457. E.R. van Dam, R.E. Kooij, The minimal spectral radius of graphs with a given diameter, Linear Algebra Appl. 423 (2007) 408–419. A. Dirac, Minimally 2-connected graphs, J. Reine Angew. Math. 228 (1967) 204–216. O. Favaron, M. Mahéo, J.-F. Saclé, Some eigenvalue properties in graphs (conjectures of Graffiti, II), Discrete Math. 111 (1993) 197–220. L. Feng, Q. Li, X. Zhang, Spectral radii of graphs with given chromatic number, Appl. Math. Lett. 20 (2007) 158–162. L. Feng, G. Yu, X. Zhang, Spectral radius of graphs with given matching number, Linear Algebra Appl. 422 (2007) 133–138. J. Guo, J. Shao, On the spectral radius of trees with fixed diameter, Linear Algebra Appl. 413 (2006) 131–147. P. Hansen, D. Stevanović, On bags and bugs, Discrete Appl. Math. 156 (2008) 986–997. H. Liu, M. Lu, On the spectral radius of quasi-tree graphs, Linear Algebra Appl. 428 (2008) 2708–2714. H. Liu, M. Lu, F. Tian, On the spectral radius of graphs with cut edges, Linear Algebra Appl. 389 (2004) 139–145. V. Nikiforov, Bounds on graph eigenvalues II, Linear Algebra Appl. 427 (2007) 183–189. V. Nikiforov, The maximum spectral radius of C4 -free graphs of given order and size, Linear Algebra Appl. 430 (2009) 2898–2905. V. Nikiforov, The spectral radius of graphs without paths and cycles of specified length, Linear Algebra Appl. 432 (2010) 2243–2256. D. Stevanović, Spectral Radius of Graphs, Academic Press, Amsterdam, 2015. D. Stevanović, M. Aouchiche, P. Hansen, On the spectral radius of graphs with a given domination number, Linear Algebra Appl. 428 (2008) 1854–1864. D. Stevanović, P. Hansen, The minimum spectral radius of graphs with a given clique number, Electron. J. Linear Algebra 17 (2008) 110–117. M. Tait, J. Tobin, Three conjectures in extremal spectral graph theory, J. Combin. Theory Ser. B 126 (2017) 137–161. D.B. West, Introduction To Graph Theory, second ed., Prentice-Hall, Upper Saddle River (NJ), 2001. B. Wu, E. Xiao, Y. Hong, The spectral radius of trees on k pendant vertices, Linear Algebra Appl. 395 (2005) 343–349. M. Xu, Y. Hong, J. Shu, M. Zhai, The minimum spectral radius of graphs with a given independence number, Linear Algebra Appl. 431 (2009) 937–945.