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On the multiplicity of α as an eigenvalue of the Aα matrix of a graph in terms of the number of pendant vertices
Linear Algebra and its Applications 594 (2020) 193–204
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Linear Algebra and its Applications www.elsevier.com/locate/laa
On the multiplicity of α as an eigenvalue of the Aα matrix of a graph in terms of the number of pendant vertices Feng Xu a , Dein Wong a,∗,1 , Fenglei Tian b a
School of Mathematics, China University of Mining and Technology, Xuzhou 221116, Jiangsu, China b School of Management, Qufu Normal University, Rizhao, 276826, Shandong, China
a r t i c l e
i n f o
Article history: Received 4 October 2019 Accepted 20 February 2020 Available online 24 February 2020 Submitted by R. Brualdi MSC: 05C50 Keywords: Aα -eigenvalues Multiplicity of an eigenvalue Signless Laplacian matrix
a b s t r a c t Let G = (V (G), E(G)) be a simple undirected graph with vertex set V (G) and edge set E(G). The cyclomatic number of a connected graph G is defined as θ(G) = |E(G)| − |V (G)| + 1. The Aα matrix of a graph G is defined by Nikiforov as Aα (G) = αD(G) + (1 − α)A(G), where α ∈ [0, 1], A(G) and D(G) respectively denotes the adjacency matrix and the diagonal matrix of the vertex degrees of G. Let mG (λ) be the multiplicity of λ as an eigenvalue of Aα (G). A cluster of G is an independent set of one or more vertices of G, each of which has the same set of neighbors. The closure of a cluster is the set of vertices of the cluster together with all the neighbors of the cluster. The degree of a cluster is the cardinality of its shared set of neighbors. A k-cluster is a cluster of degree k. The number of vertices in a k-cluster is its order. A collection of two or more k-clusters is independent if the k-clusters are pairwise disjoint and is isolated if the closures of the k-clusters are pairwise disjoint. Cardoso et al. [2] obtained a lower bound for mG (α) in terms of the number of pendant vertices. They proved that, for a simple connected graph G with p(G) > 0 pendant vertices attached at q(G) quasi-pendant vertices, mG (α) ≥ p(G) − q(G), with equality if each internal vertex is a quasi-pendant
* Corresponding author. 1
E-mail address: [email protected] (D. Wong). Supported by “the Fundamental Research Funds for the Central Universities (No. 2018ZDPY06)”.
https://doi.org/10.1016/j.laa.2020.02.025 0024-3795/© 2020 Elsevier Inc. All rights reserved.
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vertex. In this article, for a graph G of order n and with t independent k-clusters of orders r1 , . . . , rt , we prove that mG (kα) ≥ ti=1 ri − t, and if the k-clusters are isolated, then mG (kα) ≤ n − 2t, which extends Cardoso’s result to a more general case. Also, we give an upper bound for mG (α) in terms of p(G). It is proved that mG (α) ≤ p(G) +2θ(G) with θ(G) the cyclomatic number of G, the graphs G whose mG (α) attain the upper bound are characterized explicitly. © 2020 Elsevier Inc. All rights reserved.
1. Introduction The adjacency matrix A(G) = [au,v ] of G is a symmetric matrix such that au,v = 1 if u ∼ v and au,v = 0 if otherwise, where au,v is the entry of A(G) with row and column respectively indexed by u and v. The diagonal matrix of the vertex degrees of G is defined as D(G) = diag{dG (v) : v ∈ V (G)}, and the signless Laplacian matrix of G is defined as Q(G) = D(G) + A(G). In 2016, Nikiforov ([12]) merged A(G), D(G) and Q(G) into the Aα matrix as: Aα (G) = αD(G) + (1 − α)A(G), where α ∈ [0, 1] is a real number. Setting α = 0, 12 or 1, one can obtain A(G) = A0 (G); Q(G) = 2A 12 (G); and D(G) = A1 (G). Since Aα (G) is the linear combinations of A(G) and D(G), it was claimed in [12] that the Aα matrix Aα (G) can underpin a unified theory of A(G) and Q(G). Till date, there have been a few study on Aα (G), such as the characteristic polynomial ([10]), the spectral radius ([5–8,13,15,18,19]), the second largest eigenvalue ([4]), the positive semidefinitness ([14]), the number of distinct eigenvalues ([16]) and the spectrum determining problem ([9]). Among these investigations, two recent publications on multiplicity of a particular Aα -eigenvalue of a graph must be mentioned. Let mG (λ) be the multiplicity of λ as an eigenvalue of Aα (G). Cardoso et al. ([2]) proved the following result. Proposition 1.1. ([2], Inequality (3) and Corollary 3) For a simple connected graph G with p(G) > 0 pendant vertices attached at q(G) quasi-pendant vertices, mG (α) ≥ p(G)−q(G), with equality if each internal vertex is a quasi-pendant vertex.
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This result was extended to Aα -eigenvalues of signed graph by Belardo et al. ([1]). A k-cluster is a cluster of degree k. The number of vertices in a k-cluster is its order. A collection of two or more k-clusters is independent if the k-clusters are pairwise disjoint and is isolated if the closures of the k-clusters are pairwise disjoint. In this paper, we extend the above result to the case when G has some k-clusters. Theorem 1.2. Let G be a connected graph of order n and with q > 0 independent kq clusters of orders r1 , ..., rq . Then for α ∈ [0, 1), mG (kα) ≥ i=1 ri − q. Furthermore, if the k-clusters are isolated, then mG (kα) ≤ n − 2q. Remark 1. It is not difficult to see that Proposition 1.1 is a corollary of Theorem 1.2. Indeed, all pendant vertices attached at a common quasi-pendant vertex can be viewed as a 1-cluster and all 1-clusters are naturally isolated. Suppose there are q quasi-pendant vertices labeled as u1 , . . . , uq , and there are pi pendant vertices attached at ui for i = 1, . . . , q. Thus there are q 1-clusters of order p1 , . . . , pq in G. By Theorem 1.2, we have mG (α) ≥
q
pi − q = p(G) − q(G).
i=1
If each internal vertex is a quasi-pendant vertex, then n = p(G) +q(G) and thus n −2q = p(G) − q(G). By Theorem 1.2, mG (α) ≤ n − 2q = p(G) − q(G), and thus we have mG (α) = p(G) − q(G). Remark 2. If the clusters in Theorem 1.2 are assumed to be isolated then the conclusion of Theorem 1.2 can be obtained from Theorem 2.7 (i) in [3], where each cluster even has arbitrary order and degree. Belardo et al. in [2] only established a lower bound for mG (α) in terms of p(G). Thus we are motivated to give an upper bound for mG (α) in terms of p(G). The following theorem is another main result of this paper. Theorem 1.3. Let G be a connected graph with p(G) pendant vertices and with cyclomatic number θ(G). For α ∈ [0, 1), we have mG (α) ≤ p(G) + 2θ(G), the equality holds if and only if G is a cycle Cn and α =
2 cos 2πj n −1+2 cos 2πj n
with 0 < j <
n 2.
Theorem 1.3 extends the main result (Theorem 1.1) in [11] from the nullity of a graph to the eigenvalue α in Aα matrix. Applying Theorem 1.3, we immediately obtain the following corollary. Corollary 1.4. Let G be a connected graph with p(G) pendant vertices and with cyclomatic number θ(G). Then the multiplicity of eigenvalue 1 in the signless Laplacian matrix Q(G) is bounded from above by p(G) + 2θ(G), the equality holds if and only if G is a cycle Cn with n divisible by 3.
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We will prove Theorem 1.2 in Section 2 and prove Theorem 1.3 in Section 3. 2. Proof of Theorem 1.2 Some primary notations are introduced at the beginning part of this section. Throughout this paper we consider undirected simple graphs, i.e., multiple edges or loops are not permitted. The cyclomatic number of a connected graph G = (V (G), E(G)) with vertex set V (G) and edge set E(G) is defined as θ(G) = |E(G)| − |V (G)| + 1. A connected graph G is a tree (resp., a unicyclic graph; a bicyclic graph) if θ(G) = 0 (resp., θ(G) = 1; θ(G) = 2). A θ-type bicyclic graph is obtained from a pair of vertices u and v joined by three internally disjoint paths with lengths not less than one (greater than one if u and v are adjacent). The neighbor set of a vertex v in G is defined as NG (v) = {u ∈ V (G) : u ∼ v}, and the degree of v is defined as dG (v) = |NG (v)|. If dG (v) = 1, v is a pendant vertex, the vertex adjacent to a pendant vertex is a quasipendant vertex. Denote by p(G) and q(G) respectively the number of pendant vertices and the number of quasi-pendant vertices of G. A major vertex of G is a vertex of degree at least 3. A cluster of G is an independent set of one or more vertices of G, each of which has the same set of neighbors. The closure of a cluster is the set of vertices of the cluster together with all the neighbors of the cluster. The degree of a cluster is the cardinality of its shared set of neighbors. If U ⊆ V (G), we denote by G[U ] the induced subgraph of G with vertex set U , and we denote by G − U the induced subgraph of G with vertex set V (G)\U , i.e., G − U = G[V (G) \ U ]. For an induced graph H of G, we replace G − V (H) with G − H for simple. Assume the order of G is n. Since the multiplicity of α in D(G) is trivial, in the rest part of the paper, we always assume that α ∈ [0, 1). By Rn we denote the vector space of all n × 1 column vectors over the field R of all real numbers. For a vector x of Rn and a vertex v of G, we write xv as the component of x with respect to v. The characteristic subspace of Aα (G) corresponding to an eigenvalue λ is written as VGλ , i.e., VGλ = {x ∈ Rn : Aα (G)x = λx}. Obviously, mG (λ) = dim(VGλ ), since Aα (G) is symmetric and thus diagonalizable. For a subset U of V (G), we define: ZG (U ) = {x ∈ Rn : xu = 0, ∀u ∈ U }. Then ZG (U ) is a subspace of Rn with dimension n − |U |. The following lemma obtained in [17] helps us to establish an upper bound for mG (λ), especially for mG (α) and mG (kα).
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Lemma 2.1. ([17], Lemma 2.1) Let G be a graph of order n and let U be a subset of V (G). Then mG (λ) ≤ |U | if VGλ ∩ ZG (U ) = {0}. Example 2.2. As an application of this lemma, we consider Cn as an example. Arrange the vertices of Cn as Cn : v1 ∼ v2 ∼ . . . ∼ vn ∼ v1 . Let U = {v1 , v2 } and assume x ∈ ZCn (U ) ∩ VCαn , then from (2α − α)xv2 + (1 − α)(xv1 + xv3 ) = 0, we have xv3 = 0. Similarly, we have xv4 = 0. Proceeding this way, we have xv5 = . . . = xvn−1 = xvn = 0. Thus x = 0 and ZCn (U ) ∩ VCαn = {0}, which gives that mCn (α) ≤ 2. Now, we are ready to give a proof for Theorem 1.2 Proof of Theorem 1.2. Let Ω1 , . . . , Ωq be all the independent k-clusters and assume that |Ωi | = ri ≥ 2 for 1 ≤ i ≤ s and |Ωj | = rj = 1 for s + 1 ≤ j ≤ q. Label the vertices in Ωi , i = 1, . . . , q, as Ωi = {wi,1 , wi,2 , . . . , wi,ri }. For each Ωi with 1 ≤ i ≤ s, let yij , where 2 ≤ j ≤ ri , be the vector in Rn defined as ⎧ ⎪ ⎨ −1, (yij )v = 1, ⎪ ⎩ 0,
if v = wi,1 ; if v = wi,j; otherwise.
Put Y = {y1,2 . . . y1,r1 , . . . , ys,2 . . . ys,rs }. Obviously, |Y | =
s
ri − s =
i=1
q
ri − q.
i=1
It is easy to verify that all vectors in Y are linearly independent and Aα (G)y = (kα)y,
∀y ∈ Y.
That is to say y ∈ VGkα for each y ∈ Y . Hence, mG (kα) =
dim(VGkα )
≥ |Y | =
q i=1
ri − q,
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which proves the lower bound for mG (kα). Next, we prove the upper bound for mG (kα). Let Ψi be the set of k neighbors of vertices in Ωi . Since the k-clusters are isolated, Ψi ∩ Ψj = ∅ for i = j. Choose a vertex, say ui , from Ψi arbitrarily and let W = {w1,1 , u1 , . . . , wq,1 , uq }. Put U = V (G) \ W . Then |U | = n − 2q. We claim that ZG (U ) ∩ VGkα = {0}. Let x ∈ ZG (U ) ∩ VGkα . If we can prove that xv = 0 for v ∈ W , then we have x = 0 and the claim is proved. Considering the characteristic equations Aα (G)x = (kα)x at vertex wi,1 , we have (kα − kα)xwi,1 = (1 − α)
xv = (1 − α)xui ,
v∈Ψi
and thus xui = 0 for i = 1, 2, . . . , q. Now, we have xv = 0 for v ∈ / {w1,1 , . . . , wq,1 }. Finally, in order to prove xv = 0 for v ∈ {w1,1 . . . , wq,1 }, we consider the characteristic equations Aα (G)x = (kα)x at vertex ui . Then we have (kα − dG (ui )α)xui = (1 − α)
xv = (1 − α)xwi,1 ,
v∼ui
which gives that xwi,1 = 0 for 1 ≤ i ≤ q. Therefore, ZG (U ) ∩ VGkα = {0}. By Lemma 2.1, we have mG (kα) ≤ |U | = n − 2q, which completes the proof. 3. Proof of Theorem 1.3 Now, we focus on the eigenvalue α of AG (α). The following lemma is key for the proof of Theorem 1.3. Lemma 3.1. Let T be a vertex cut set of G with H a connected component of G − T . If a subset S of V (H) ∪ T satisfies ZG (S) ∩ VGα ⊆ ZG (V (H) ∪ T ), then mG (α) ≤ |S| + mG−H (α). Further, if there is a subset Q of V (G) which is disjoint with T ∪ V (H) α such that ZG−H (T ∪ Q) ∩ VG−H = {0}, then mG (α) ≤ |S| + |Q|. Proof. Let K be the induced subgraph with vertex set V (G) \ (V (H) ∪ T ). Arrange the vertices of G such that ⎡ ⎤ 0 AH B ⎢ ⎥ Aα (G) = ⎣ B t AT C ⎦, 0 C t AK where AH , AT , AK are respectively the major submatrix of Aα (G) corresponding to H, G[T ] and K and B t is the transpose of B.
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α We claim that x|G−H ∈ VG−H if x ∈ ZG (S) ∩ VGα . Assume that x ∈ ZG (S) ∩ VGα . We have xv = 0 for v ∈ ZG (V (H) ∪ T ), since ZG (S) ∩ VGα ⊆ ZG (V (H) ∪ T ). From Aα (G)x = αx it follows that
AT Ct
C AK
0 x|K
=α
0 x|K
.
Thus we have Aα (G − H)
0 x|K
=α
0 x|K
,
AT − αD C since Aα (G − H) = , where D = diag{dH (t) : t ∈ T } is a diagonal AK Ct matrix and dH (t) is the number of vertices of H adjacent to t. This completes the proof for the claim. Suppose the order of G and G − H are respectively n and m. Let ϕ be the linear mapping from Rn to Rm defined as ϕ(x) = x|G−H for x ∈ Rn . The claim has proved that α ϕ(ZG (S) ∩ VGα ) ⊆ VG−H .
It is easy to see that x = 0 if x ∈ ZG (S) ∩ VGα such that ϕ(x) = 0. Thus the restriction of ϕ to ZG (S) ∩ VGα is injective. Hence, α dim(ZG (S) ∩ VGα ) ≤ dim(VG−H ),
which follows that α dim(ZG (S)) + dim(VGα ) − n ≤ dim(VG−H ).
Recalling that dim(ZG (S)) = n − |S|, we have mG (α) ≤ |S| + mG−H (α), which proves the first assertion. For the second assertion, it suffices to prove that ZG (S ∪ Q) ∩ VGα = {0} (in view of Lemma 2.1). Assume that x ∈ ZG (S ∪ Q) ∩ VGα . Thus x ∈ ZG (T ). Further, x|G−H ∈ α ZG−H (T ∪ Q). From the first assertion, x|G−H ∈ VG−H . The condition for Q implies that x|G−H = 0. As x ∈ ZG (V (H) ∪ T ), x|H = 0. Hence x = 0, and we are done. Applying Lemma 3.1, we can bound mG (α) for some special graphs.
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Lemma 3.2. Let G be a graph obtained from a path Pl with l ≥ 2 and a disjoint graph H by identifying a pendant vertex of Pl with a vertex of H. Then mG (α) ≤ mH (α) + 1. Moreover, if H is a cycle, then mG (α) ≤ 2. Proof. Arrange the vertices of Pl as Pl : v1 ∼ v2 ∼ . . . ∼ vl with vl shared by H. Let T = {vl }, a cut set of G, and let S = {v1 }. If x ∈ ZG (S) ∩ VGα , then from (α − α)xv1 + (1 − α)xv2 = 0 and (2α − α)xvi + (1 − α)(xvi−1 + xvi+1 ) = 0, i = 2, . . . , l − 1, we have xv2 = xv3 = . . . = xvl = 0 in turn. Thus x ∈ ZG (V (Pl )) and ZG (S) ∩ VGα ⊆ ZG (V (Pl )). Applying Lemma 3.1, we have mG (α) ≤ mH (α) + 1. For the second assertion, let Cm : u1 ∼ u2 ∼ . . . ∼ um ∼ u1 , where u1 = vl . Let Q = {u2 }. Example 2.2 has proven that ZCn (T ∪ Q) ∩ VCαn = {0}. By Lemma 3.1, we have mG (α) ≤ 2. Lemma 3.3. Let G be a graph obtained from a cycle Cs and a disjoint graph H which are joined by a path Pm with m ≥ 1, i.e., identify one pendant vertex of Pm with a vertex of Cs and another with a vertex of H. Then mG (α) ≤ mH (α) + 2. Moreover, if H is a cycle, then mG (α) ≤ 3. Proof. Assume that m ≥ 2 (the case when m = 1 can be similarly proved). Suppose the vertices of Cs and Pm are respectively arranged as Cs : v1 ∼ v2 ∼ . . . ∼ vs ∼ v1 ; Pm : u1 ∼ u2 ∼ . . . ∼ um , where v1 = u1 and um is also a vertex of H. Let T = {um } and let S = {v1 , v2 }. If x ∈ ZG (S) ∩ VGα , then from (2α − α)xvi + (1 − α)(xvi−1 + xvi+1 ) = 0, i = 2, . . . , s − 1, we have xv3 = xv4 = . . . = xvs−1 = xvs = 0
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in turn. Furthermore, it follows from (2α − α)xui + (1 − α)(xui−1 + xui+1 ) = 0, i = 2, . . . , m − 1, we have xu 2 = xu 3 = . . . = xu m = 0 in turn. Hence, ZG (S) ∩ VGα ⊆ ZG (V (G1 ) ∪ T ), where G1 is the component of G − um containing Cs . Applying Lemma 3.1, we have mG (α) ≤ mH (α) + 2. For the second assertion, put Q to be the set consisting a vertex in H = Ct which is adjacent to um . In view of Example 2.2, ZCt (T ∪ Q) ∩ VCαt = {0}. Thus, we have mG (α) ≤ |S| + |Q| = 3. Lemma 3.4. Let G be a simple graph obtained from a graph H and a disjoint path Pm with m ≥ 2 by identifying two pendant vertices of Pm respectively with two distinct vertices u, v of H. Then mG (α) ≤ mH (α) +2. Moreover, if G is the θ-type bicyclic graph obtained from a pair of vertices u and v, joined by three internally disjoint paths Pk , Pl and Pm with k ≥ l ≥ m ≥ 2, then mG (α) ≤ 3. Proof. To guarantee G to be simple, m must be greater than 2 if u, v are adjacent in H. Arrange the vertices of Pm as Pm : v = v1 ∼ v2 ∼ . . . ∼ vm = u, where v, u are also vertices of H. Let T = {u, v}. Then G − T has a component Pm−2 for m ≥ 3. Put S = {v1 , v2 }. If x ∈ ZG (S) ∩ VGα , then from (2α − α)xvi + (1 − α)(xvi−1 + xvi+1 ) = 0, i = 2, . . . , m − 1, we have xvi = 0 for i = 1, 2, . . . , m. Thus, ZG (S) ∩ VGα ⊆ ZG (V (Pm−2 ) ∪ T ) = ZG (V (Pm )), even if m = 2. Applying Lemma 3.1, we have mG (α) ≤ mH (α) + 2. The proof for the second assertion is similar as in Lemma 3.3 and is omitted. Now, we are ready to give a proof for Theorem 1.3. Proof of Theorem 1.3. Firstly, we proceed by induction on the order n of G to prove the following claim. • If G is a connected graph that is not a cycle, then mG (α) ≤ 2θ(G) + p(G) − 1.
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If n = 2, the claim is trivial. Assume the claim holds for connected graphs of order at most n − 1 and let G be a connected graph of order n ≥ 3 that is not a cycle. We need to consider three cases. Case 1 G has a pendant vertex v. If G has no major vertex, then G is a path and the claim follows from the observation that mG (α) ≤ 1. Suppose G has major vertices. Let u be the major vertex such that m = dG (v, u) ≤ dG (v, w) for all major vertices w of G. Let H be the induced subgraph of G obtained from G by deleting the path Pm+1 joining v and u. Then G is a graph obtained from H := H + u and Pm+1 by identifying one pendant vertex of Pm+1 with u of H. If H is a cycle then by Lemma 3.2, mG (α) ≤ 2 = 2θ(G) + p(G) − 1, as required. If H is not a cycle, the induction hypothesis implies that mH (α) ≤ 2θ(H) + p(H) − 1. Noting that θ(H) = θ(G) and p(H) = p(G) − 1 and applying Lemma 3.2, we have mG (α) ≤ mH (α) + 1 ≤ (2θ(H) + p(H) − 1) + 1 = 2θ(G) + p(G) − 1. Case 2 G has no pendant vertices and distinct cycles of G have no common edges. Let Cm be a pendant cycle of G, namely it has only one major vertex v of G, and let H = G − Cm + v. If H is a cycle, then we are done by Lemma 3.3. Suppose that H is not a cycle and thus mH (α) ≤ 2θ(H) + p(H) − 1, by the induction hypothesis. If dG (v) ≥ 4, then p(H) = 0. Applying Lemma 3.3 and noting that θ(H) = θ(G) − 1 we have the required result by mG (α) ≤ mH (α) + 2 ≤ (2θ(H) − 1) + 2 = 2θ(G) − 1. Suppose that dG (v) = 3. Obviously, G has at least two major vertices (otherwise, G has pendant vertices). Let u be the major vertex of G which minimizes dG (v, u). Let H be the induced subgraph of G obtained by deleting the pendant cycle Cm together with the path from v to u. Then G can be viewed as the graph obtained from Cm and H := H +u joined by the path with two pendant vertices v and u. Applying Lemma 3.3, we have mG (α) ≤ mH (α) + 2. Noting that θ(H) = θ(G) − 1, p(H) = p(G) = 0 and mH (α) ≤ 2θ(H) − 1 (using the induction hypothesis), we have mG (α) ≤ mH (α) + 2 ≤ 2θ(H) − 1 + 2 ≤ 2θ(G) − 1. Case 3 G has no pendant vertices and a pair of cycles of G have common edges. Let e = u v be a common edge of two cycles Ck and Cl . Let u (resp., v) be a major vertex of G which minimizes dG−e (u , u) (resp., dG−e (v , v)). It should be indicated that u = u if u itself is a major vertex. Let P be the path beginning at u, across u , v , and ending at v and let H = G −P , where P = P −u −v. Then G is the graph obtained from H and P by identifying two pendant vertices of P respectively with two vertices u, v of H. If H is a cycle, then G is a θ-type bicyclic graph and we are done by Lemma 3.4. Suppose
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that H is not a cycle. Then mH (α) ≤ 2θ(H) − 1 (using the induction hypothesis). By Lemma 3.4, mG (α) ≤ mH (α) + 2. Noting that θ(G) = θ(H) + 1, p(G) = p(H) = 0, we have mG (α) ≤ mH (α) + 2 ≤ 2θ(H) − 1 + 2 ≤ 2θ(G) − 1. The proof for the claim is completed. If G is a cycle, the example has proven mG (α) ≤ 2 = 2θ(G) + p(G). Therefore, mG (α) ≤ 2 = 2θ(G) + p(G) for all connected graph G. If mG (α) = 2θ(G)+p(G), then the claim just proved (that is mG (α) ≤ 2θ(G)+p(G)−1 if G is a connected graph that is not a cycle) indicates that G is a cycle. Since the Aα spectrum of a cycle Cn is {2α+2(1 −α) cos 2πj n : j = 0, 1, . . . , n −1}, we have mCn (α) = 2 if and only if α =
2 cos 2πj n −1+2 cos 2πj n
with 0 < j <
n 2.
Declaration of competing interest There is no competing interest. References [1] F. Belardo, M. Brunetti, A. Ciampella, On the multiplicity of α as an Aα (Γ)-eigenvalue of signed graphs with pendant vertices, Discrete Math. 342 (2019) 2223–2233. [2] D.M. Cardoso, G. Pastén, O. Rojo, On the multiplicity of α as an eigenvalue of Aα (G) of graphs with pendant vertices, Linear Algebra Appl. 552 (2018) 52–70. [3] D.M. Cardoso, G. Pastén, O. Rojo, Graphs with clusters perturbed by regular graphs-Aα -spectrum and applications, Discuss. Math., Graph Theory 40 (2020) 451–466. [4] Y. Chen, D. Li, J. Meng, On the second largest Aα -eigenvalues of graphs, Linear Algebra Appl. 580 (2019) 343–358. [5] Y. Chen, D. Li, Z. Wang, J. Meng, Aα -spectral radius of the second power of a graph, Appl. Math. Comput. 359 (2019) 418–425. [6] H. Guo, B. Zhou, On the α-spectral radius of graphs, preprint, arXiv:1805.03456. [7] D. Li, Y. Chen, J. Meng, The Aα -spectral radius of graphs with given degree sequence, preprint, arXiv:1806.02603. [8] H. Lin, X. Huang, J. Xue, A note on the Aα -spectral radius of graphs, Linear Algebra Appl. 557 (2018) 430–437. [9] H. Lin, X. Liu, J. Xue, Graphs determined by their Aα -spectra, Discrete Math. 342 (2019) 441–450. [10] X. Liu, S. Liu, On the Aα -characteristic polynomial of a graph, Linear Algebra Appl. 546 (2018) 274–288. [11] X. Ma, D. Wong, F. Tian, Nullity of a graph in terms of the dimension of cycle space and the number of pendant vertices, Discrete Appl. Math. 215 (2016) 171–176. [12] V. Nikiforov, Merging the A- and Q-spectral theories, Appl. Anal. Discrete Math. 11 (2017) 81–107. [13] V. Nikiforov, G. Pastén, O. Rojo, R.L. Soto, On the Aα -spectra of trees, Linear Algebra Appl. 520 (2017) 286–305. [14] V. Nikiforov, O. Rojo, A note on the positive semidefinitness of Aα (G), Linear Algebra Appl. 519 (2017) 156–163. [15] V. Nikiforov, O. Rojo, On the α-index of graphs with pendent paths, Linear Algebra Appl. 550 (2018) 87–104. [16] M.A. Tahir, X. Zhang, Graphs with three distinct α-eigenvalues, Acta Math. Vietnam. 43 (2018) 649–659. [17] L. Wang, X. Fang, X. Geng, F. Tian, On the multiplicity of an arbitrary Aα -eigenvalue of a connected graph, Linear Algebra Appl. 589 (2020) 28–38.
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[18] J. Xue, H. Lin, S. Liu, J. Shu, On the Aα -spectral radius of a graph, Linear Algebra Appl. 550 (2018) 105–120. [19] Z. Yu, L. Kang, L. Liu, E. Shan, The extremal α-index of outerplanar and planar graphs, Appl. Math. Comput. 343 (2019) 90–99.
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Linear Algebra and its Applications www.elsevier.com/locate/laa
On the multiplicity of α as an eigenvalue of the Aα matrix of a graph in terms of the number of pendant vertices Feng Xu a , Dein Wong a,∗,1 , Fenglei Tian b a
School of Mathematics, China University of Mining and Technology, Xuzhou 221116, Jiangsu, China b School of Management, Qufu Normal University, Rizhao, 276826, Shandong, China
a r t i c l e
i n f o
Article history: Received 4 October 2019 Accepted 20 February 2020 Available online 24 February 2020 Submitted by R. Brualdi MSC: 05C50 Keywords: Aα -eigenvalues Multiplicity of an eigenvalue Signless Laplacian matrix
a b s t r a c t Let G = (V (G), E(G)) be a simple undirected graph with vertex set V (G) and edge set E(G). The cyclomatic number of a connected graph G is defined as θ(G) = |E(G)| − |V (G)| + 1. The Aα matrix of a graph G is defined by Nikiforov as Aα (G) = αD(G) + (1 − α)A(G), where α ∈ [0, 1], A(G) and D(G) respectively denotes the adjacency matrix and the diagonal matrix of the vertex degrees of G. Let mG (λ) be the multiplicity of λ as an eigenvalue of Aα (G). A cluster of G is an independent set of one or more vertices of G, each of which has the same set of neighbors. The closure of a cluster is the set of vertices of the cluster together with all the neighbors of the cluster. The degree of a cluster is the cardinality of its shared set of neighbors. A k-cluster is a cluster of degree k. The number of vertices in a k-cluster is its order. A collection of two or more k-clusters is independent if the k-clusters are pairwise disjoint and is isolated if the closures of the k-clusters are pairwise disjoint. Cardoso et al. [2] obtained a lower bound for mG (α) in terms of the number of pendant vertices. They proved that, for a simple connected graph G with p(G) > 0 pendant vertices attached at q(G) quasi-pendant vertices, mG (α) ≥ p(G) − q(G), with equality if each internal vertex is a quasi-pendant
* Corresponding author. 1
E-mail address: [email protected] (D. Wong). Supported by “the Fundamental Research Funds for the Central Universities (No. 2018ZDPY06)”.
https://doi.org/10.1016/j.laa.2020.02.025 0024-3795/© 2020 Elsevier Inc. All rights reserved.
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vertex. In this article, for a graph G of order n and with t independent k-clusters of orders r1 , . . . , rt , we prove that mG (kα) ≥ ti=1 ri − t, and if the k-clusters are isolated, then mG (kα) ≤ n − 2t, which extends Cardoso’s result to a more general case. Also, we give an upper bound for mG (α) in terms of p(G). It is proved that mG (α) ≤ p(G) +2θ(G) with θ(G) the cyclomatic number of G, the graphs G whose mG (α) attain the upper bound are characterized explicitly. © 2020 Elsevier Inc. All rights reserved.
1. Introduction The adjacency matrix A(G) = [au,v ] of G is a symmetric matrix such that au,v = 1 if u ∼ v and au,v = 0 if otherwise, where au,v is the entry of A(G) with row and column respectively indexed by u and v. The diagonal matrix of the vertex degrees of G is defined as D(G) = diag{dG (v) : v ∈ V (G)}, and the signless Laplacian matrix of G is defined as Q(G) = D(G) + A(G). In 2016, Nikiforov ([12]) merged A(G), D(G) and Q(G) into the Aα matrix as: Aα (G) = αD(G) + (1 − α)A(G), where α ∈ [0, 1] is a real number. Setting α = 0, 12 or 1, one can obtain A(G) = A0 (G); Q(G) = 2A 12 (G); and D(G) = A1 (G). Since Aα (G) is the linear combinations of A(G) and D(G), it was claimed in [12] that the Aα matrix Aα (G) can underpin a unified theory of A(G) and Q(G). Till date, there have been a few study on Aα (G), such as the characteristic polynomial ([10]), the spectral radius ([5–8,13,15,18,19]), the second largest eigenvalue ([4]), the positive semidefinitness ([14]), the number of distinct eigenvalues ([16]) and the spectrum determining problem ([9]). Among these investigations, two recent publications on multiplicity of a particular Aα -eigenvalue of a graph must be mentioned. Let mG (λ) be the multiplicity of λ as an eigenvalue of Aα (G). Cardoso et al. ([2]) proved the following result. Proposition 1.1. ([2], Inequality (3) and Corollary 3) For a simple connected graph G with p(G) > 0 pendant vertices attached at q(G) quasi-pendant vertices, mG (α) ≥ p(G)−q(G), with equality if each internal vertex is a quasi-pendant vertex.
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This result was extended to Aα -eigenvalues of signed graph by Belardo et al. ([1]). A k-cluster is a cluster of degree k. The number of vertices in a k-cluster is its order. A collection of two or more k-clusters is independent if the k-clusters are pairwise disjoint and is isolated if the closures of the k-clusters are pairwise disjoint. In this paper, we extend the above result to the case when G has some k-clusters. Theorem 1.2. Let G be a connected graph of order n and with q > 0 independent kq clusters of orders r1 , ..., rq . Then for α ∈ [0, 1), mG (kα) ≥ i=1 ri − q. Furthermore, if the k-clusters are isolated, then mG (kα) ≤ n − 2q. Remark 1. It is not difficult to see that Proposition 1.1 is a corollary of Theorem 1.2. Indeed, all pendant vertices attached at a common quasi-pendant vertex can be viewed as a 1-cluster and all 1-clusters are naturally isolated. Suppose there are q quasi-pendant vertices labeled as u1 , . . . , uq , and there are pi pendant vertices attached at ui for i = 1, . . . , q. Thus there are q 1-clusters of order p1 , . . . , pq in G. By Theorem 1.2, we have mG (α) ≥
q
pi − q = p(G) − q(G).
i=1
If each internal vertex is a quasi-pendant vertex, then n = p(G) +q(G) and thus n −2q = p(G) − q(G). By Theorem 1.2, mG (α) ≤ n − 2q = p(G) − q(G), and thus we have mG (α) = p(G) − q(G). Remark 2. If the clusters in Theorem 1.2 are assumed to be isolated then the conclusion of Theorem 1.2 can be obtained from Theorem 2.7 (i) in [3], where each cluster even has arbitrary order and degree. Belardo et al. in [2] only established a lower bound for mG (α) in terms of p(G). Thus we are motivated to give an upper bound for mG (α) in terms of p(G). The following theorem is another main result of this paper. Theorem 1.3. Let G be a connected graph with p(G) pendant vertices and with cyclomatic number θ(G). For α ∈ [0, 1), we have mG (α) ≤ p(G) + 2θ(G), the equality holds if and only if G is a cycle Cn and α =
2 cos 2πj n −1+2 cos 2πj n
with 0 < j <
n 2.
Theorem 1.3 extends the main result (Theorem 1.1) in [11] from the nullity of a graph to the eigenvalue α in Aα matrix. Applying Theorem 1.3, we immediately obtain the following corollary. Corollary 1.4. Let G be a connected graph with p(G) pendant vertices and with cyclomatic number θ(G). Then the multiplicity of eigenvalue 1 in the signless Laplacian matrix Q(G) is bounded from above by p(G) + 2θ(G), the equality holds if and only if G is a cycle Cn with n divisible by 3.
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We will prove Theorem 1.2 in Section 2 and prove Theorem 1.3 in Section 3. 2. Proof of Theorem 1.2 Some primary notations are introduced at the beginning part of this section. Throughout this paper we consider undirected simple graphs, i.e., multiple edges or loops are not permitted. The cyclomatic number of a connected graph G = (V (G), E(G)) with vertex set V (G) and edge set E(G) is defined as θ(G) = |E(G)| − |V (G)| + 1. A connected graph G is a tree (resp., a unicyclic graph; a bicyclic graph) if θ(G) = 0 (resp., θ(G) = 1; θ(G) = 2). A θ-type bicyclic graph is obtained from a pair of vertices u and v joined by three internally disjoint paths with lengths not less than one (greater than one if u and v are adjacent). The neighbor set of a vertex v in G is defined as NG (v) = {u ∈ V (G) : u ∼ v}, and the degree of v is defined as dG (v) = |NG (v)|. If dG (v) = 1, v is a pendant vertex, the vertex adjacent to a pendant vertex is a quasipendant vertex. Denote by p(G) and q(G) respectively the number of pendant vertices and the number of quasi-pendant vertices of G. A major vertex of G is a vertex of degree at least 3. A cluster of G is an independent set of one or more vertices of G, each of which has the same set of neighbors. The closure of a cluster is the set of vertices of the cluster together with all the neighbors of the cluster. The degree of a cluster is the cardinality of its shared set of neighbors. If U ⊆ V (G), we denote by G[U ] the induced subgraph of G with vertex set U , and we denote by G − U the induced subgraph of G with vertex set V (G)\U , i.e., G − U = G[V (G) \ U ]. For an induced graph H of G, we replace G − V (H) with G − H for simple. Assume the order of G is n. Since the multiplicity of α in D(G) is trivial, in the rest part of the paper, we always assume that α ∈ [0, 1). By Rn we denote the vector space of all n × 1 column vectors over the field R of all real numbers. For a vector x of Rn and a vertex v of G, we write xv as the component of x with respect to v. The characteristic subspace of Aα (G) corresponding to an eigenvalue λ is written as VGλ , i.e., VGλ = {x ∈ Rn : Aα (G)x = λx}. Obviously, mG (λ) = dim(VGλ ), since Aα (G) is symmetric and thus diagonalizable. For a subset U of V (G), we define: ZG (U ) = {x ∈ Rn : xu = 0, ∀u ∈ U }. Then ZG (U ) is a subspace of Rn with dimension n − |U |. The following lemma obtained in [17] helps us to establish an upper bound for mG (λ), especially for mG (α) and mG (kα).
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Lemma 2.1. ([17], Lemma 2.1) Let G be a graph of order n and let U be a subset of V (G). Then mG (λ) ≤ |U | if VGλ ∩ ZG (U ) = {0}. Example 2.2. As an application of this lemma, we consider Cn as an example. Arrange the vertices of Cn as Cn : v1 ∼ v2 ∼ . . . ∼ vn ∼ v1 . Let U = {v1 , v2 } and assume x ∈ ZCn (U ) ∩ VCαn , then from (2α − α)xv2 + (1 − α)(xv1 + xv3 ) = 0, we have xv3 = 0. Similarly, we have xv4 = 0. Proceeding this way, we have xv5 = . . . = xvn−1 = xvn = 0. Thus x = 0 and ZCn (U ) ∩ VCαn = {0}, which gives that mCn (α) ≤ 2. Now, we are ready to give a proof for Theorem 1.2 Proof of Theorem 1.2. Let Ω1 , . . . , Ωq be all the independent k-clusters and assume that |Ωi | = ri ≥ 2 for 1 ≤ i ≤ s and |Ωj | = rj = 1 for s + 1 ≤ j ≤ q. Label the vertices in Ωi , i = 1, . . . , q, as Ωi = {wi,1 , wi,2 , . . . , wi,ri }. For each Ωi with 1 ≤ i ≤ s, let yij , where 2 ≤ j ≤ ri , be the vector in Rn defined as ⎧ ⎪ ⎨ −1, (yij )v = 1, ⎪ ⎩ 0,
if v = wi,1 ; if v = wi,j; otherwise.
Put Y = {y1,2 . . . y1,r1 , . . . , ys,2 . . . ys,rs }. Obviously, |Y | =
s
ri − s =
i=1
q
ri − q.
i=1
It is easy to verify that all vectors in Y are linearly independent and Aα (G)y = (kα)y,
∀y ∈ Y.
That is to say y ∈ VGkα for each y ∈ Y . Hence, mG (kα) =
dim(VGkα )
≥ |Y | =
q i=1
ri − q,
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which proves the lower bound for mG (kα). Next, we prove the upper bound for mG (kα). Let Ψi be the set of k neighbors of vertices in Ωi . Since the k-clusters are isolated, Ψi ∩ Ψj = ∅ for i = j. Choose a vertex, say ui , from Ψi arbitrarily and let W = {w1,1 , u1 , . . . , wq,1 , uq }. Put U = V (G) \ W . Then |U | = n − 2q. We claim that ZG (U ) ∩ VGkα = {0}. Let x ∈ ZG (U ) ∩ VGkα . If we can prove that xv = 0 for v ∈ W , then we have x = 0 and the claim is proved. Considering the characteristic equations Aα (G)x = (kα)x at vertex wi,1 , we have (kα − kα)xwi,1 = (1 − α)
xv = (1 − α)xui ,
v∈Ψi
and thus xui = 0 for i = 1, 2, . . . , q. Now, we have xv = 0 for v ∈ / {w1,1 , . . . , wq,1 }. Finally, in order to prove xv = 0 for v ∈ {w1,1 . . . , wq,1 }, we consider the characteristic equations Aα (G)x = (kα)x at vertex ui . Then we have (kα − dG (ui )α)xui = (1 − α)
xv = (1 − α)xwi,1 ,
v∼ui
which gives that xwi,1 = 0 for 1 ≤ i ≤ q. Therefore, ZG (U ) ∩ VGkα = {0}. By Lemma 2.1, we have mG (kα) ≤ |U | = n − 2q, which completes the proof. 3. Proof of Theorem 1.3 Now, we focus on the eigenvalue α of AG (α). The following lemma is key for the proof of Theorem 1.3. Lemma 3.1. Let T be a vertex cut set of G with H a connected component of G − T . If a subset S of V (H) ∪ T satisfies ZG (S) ∩ VGα ⊆ ZG (V (H) ∪ T ), then mG (α) ≤ |S| + mG−H (α). Further, if there is a subset Q of V (G) which is disjoint with T ∪ V (H) α such that ZG−H (T ∪ Q) ∩ VG−H = {0}, then mG (α) ≤ |S| + |Q|. Proof. Let K be the induced subgraph with vertex set V (G) \ (V (H) ∪ T ). Arrange the vertices of G such that ⎡ ⎤ 0 AH B ⎢ ⎥ Aα (G) = ⎣ B t AT C ⎦, 0 C t AK where AH , AT , AK are respectively the major submatrix of Aα (G) corresponding to H, G[T ] and K and B t is the transpose of B.
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α We claim that x|G−H ∈ VG−H if x ∈ ZG (S) ∩ VGα . Assume that x ∈ ZG (S) ∩ VGα . We have xv = 0 for v ∈ ZG (V (H) ∪ T ), since ZG (S) ∩ VGα ⊆ ZG (V (H) ∪ T ). From Aα (G)x = αx it follows that
AT Ct
C AK
0 x|K
=α
0 x|K
.
Thus we have Aα (G − H)
0 x|K
=α
0 x|K
,
AT − αD C since Aα (G − H) = , where D = diag{dH (t) : t ∈ T } is a diagonal AK Ct matrix and dH (t) is the number of vertices of H adjacent to t. This completes the proof for the claim. Suppose the order of G and G − H are respectively n and m. Let ϕ be the linear mapping from Rn to Rm defined as ϕ(x) = x|G−H for x ∈ Rn . The claim has proved that α ϕ(ZG (S) ∩ VGα ) ⊆ VG−H .
It is easy to see that x = 0 if x ∈ ZG (S) ∩ VGα such that ϕ(x) = 0. Thus the restriction of ϕ to ZG (S) ∩ VGα is injective. Hence, α dim(ZG (S) ∩ VGα ) ≤ dim(VG−H ),
which follows that α dim(ZG (S)) + dim(VGα ) − n ≤ dim(VG−H ).
Recalling that dim(ZG (S)) = n − |S|, we have mG (α) ≤ |S| + mG−H (α), which proves the first assertion. For the second assertion, it suffices to prove that ZG (S ∪ Q) ∩ VGα = {0} (in view of Lemma 2.1). Assume that x ∈ ZG (S ∪ Q) ∩ VGα . Thus x ∈ ZG (T ). Further, x|G−H ∈ α ZG−H (T ∪ Q). From the first assertion, x|G−H ∈ VG−H . The condition for Q implies that x|G−H = 0. As x ∈ ZG (V (H) ∪ T ), x|H = 0. Hence x = 0, and we are done. Applying Lemma 3.1, we can bound mG (α) for some special graphs.
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Lemma 3.2. Let G be a graph obtained from a path Pl with l ≥ 2 and a disjoint graph H by identifying a pendant vertex of Pl with a vertex of H. Then mG (α) ≤ mH (α) + 1. Moreover, if H is a cycle, then mG (α) ≤ 2. Proof. Arrange the vertices of Pl as Pl : v1 ∼ v2 ∼ . . . ∼ vl with vl shared by H. Let T = {vl }, a cut set of G, and let S = {v1 }. If x ∈ ZG (S) ∩ VGα , then from (α − α)xv1 + (1 − α)xv2 = 0 and (2α − α)xvi + (1 − α)(xvi−1 + xvi+1 ) = 0, i = 2, . . . , l − 1, we have xv2 = xv3 = . . . = xvl = 0 in turn. Thus x ∈ ZG (V (Pl )) and ZG (S) ∩ VGα ⊆ ZG (V (Pl )). Applying Lemma 3.1, we have mG (α) ≤ mH (α) + 1. For the second assertion, let Cm : u1 ∼ u2 ∼ . . . ∼ um ∼ u1 , where u1 = vl . Let Q = {u2 }. Example 2.2 has proven that ZCn (T ∪ Q) ∩ VCαn = {0}. By Lemma 3.1, we have mG (α) ≤ 2. Lemma 3.3. Let G be a graph obtained from a cycle Cs and a disjoint graph H which are joined by a path Pm with m ≥ 1, i.e., identify one pendant vertex of Pm with a vertex of Cs and another with a vertex of H. Then mG (α) ≤ mH (α) + 2. Moreover, if H is a cycle, then mG (α) ≤ 3. Proof. Assume that m ≥ 2 (the case when m = 1 can be similarly proved). Suppose the vertices of Cs and Pm are respectively arranged as Cs : v1 ∼ v2 ∼ . . . ∼ vs ∼ v1 ; Pm : u1 ∼ u2 ∼ . . . ∼ um , where v1 = u1 and um is also a vertex of H. Let T = {um } and let S = {v1 , v2 }. If x ∈ ZG (S) ∩ VGα , then from (2α − α)xvi + (1 − α)(xvi−1 + xvi+1 ) = 0, i = 2, . . . , s − 1, we have xv3 = xv4 = . . . = xvs−1 = xvs = 0
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in turn. Furthermore, it follows from (2α − α)xui + (1 − α)(xui−1 + xui+1 ) = 0, i = 2, . . . , m − 1, we have xu 2 = xu 3 = . . . = xu m = 0 in turn. Hence, ZG (S) ∩ VGα ⊆ ZG (V (G1 ) ∪ T ), where G1 is the component of G − um containing Cs . Applying Lemma 3.1, we have mG (α) ≤ mH (α) + 2. For the second assertion, put Q to be the set consisting a vertex in H = Ct which is adjacent to um . In view of Example 2.2, ZCt (T ∪ Q) ∩ VCαt = {0}. Thus, we have mG (α) ≤ |S| + |Q| = 3. Lemma 3.4. Let G be a simple graph obtained from a graph H and a disjoint path Pm with m ≥ 2 by identifying two pendant vertices of Pm respectively with two distinct vertices u, v of H. Then mG (α) ≤ mH (α) +2. Moreover, if G is the θ-type bicyclic graph obtained from a pair of vertices u and v, joined by three internally disjoint paths Pk , Pl and Pm with k ≥ l ≥ m ≥ 2, then mG (α) ≤ 3. Proof. To guarantee G to be simple, m must be greater than 2 if u, v are adjacent in H. Arrange the vertices of Pm as Pm : v = v1 ∼ v2 ∼ . . . ∼ vm = u, where v, u are also vertices of H. Let T = {u, v}. Then G − T has a component Pm−2 for m ≥ 3. Put S = {v1 , v2 }. If x ∈ ZG (S) ∩ VGα , then from (2α − α)xvi + (1 − α)(xvi−1 + xvi+1 ) = 0, i = 2, . . . , m − 1, we have xvi = 0 for i = 1, 2, . . . , m. Thus, ZG (S) ∩ VGα ⊆ ZG (V (Pm−2 ) ∪ T ) = ZG (V (Pm )), even if m = 2. Applying Lemma 3.1, we have mG (α) ≤ mH (α) + 2. The proof for the second assertion is similar as in Lemma 3.3 and is omitted. Now, we are ready to give a proof for Theorem 1.3. Proof of Theorem 1.3. Firstly, we proceed by induction on the order n of G to prove the following claim. • If G is a connected graph that is not a cycle, then mG (α) ≤ 2θ(G) + p(G) − 1.
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If n = 2, the claim is trivial. Assume the claim holds for connected graphs of order at most n − 1 and let G be a connected graph of order n ≥ 3 that is not a cycle. We need to consider three cases. Case 1 G has a pendant vertex v. If G has no major vertex, then G is a path and the claim follows from the observation that mG (α) ≤ 1. Suppose G has major vertices. Let u be the major vertex such that m = dG (v, u) ≤ dG (v, w) for all major vertices w of G. Let H be the induced subgraph of G obtained from G by deleting the path Pm+1 joining v and u. Then G is a graph obtained from H := H + u and Pm+1 by identifying one pendant vertex of Pm+1 with u of H. If H is a cycle then by Lemma 3.2, mG (α) ≤ 2 = 2θ(G) + p(G) − 1, as required. If H is not a cycle, the induction hypothesis implies that mH (α) ≤ 2θ(H) + p(H) − 1. Noting that θ(H) = θ(G) and p(H) = p(G) − 1 and applying Lemma 3.2, we have mG (α) ≤ mH (α) + 1 ≤ (2θ(H) + p(H) − 1) + 1 = 2θ(G) + p(G) − 1. Case 2 G has no pendant vertices and distinct cycles of G have no common edges. Let Cm be a pendant cycle of G, namely it has only one major vertex v of G, and let H = G − Cm + v. If H is a cycle, then we are done by Lemma 3.3. Suppose that H is not a cycle and thus mH (α) ≤ 2θ(H) + p(H) − 1, by the induction hypothesis. If dG (v) ≥ 4, then p(H) = 0. Applying Lemma 3.3 and noting that θ(H) = θ(G) − 1 we have the required result by mG (α) ≤ mH (α) + 2 ≤ (2θ(H) − 1) + 2 = 2θ(G) − 1. Suppose that dG (v) = 3. Obviously, G has at least two major vertices (otherwise, G has pendant vertices). Let u be the major vertex of G which minimizes dG (v, u). Let H be the induced subgraph of G obtained by deleting the pendant cycle Cm together with the path from v to u. Then G can be viewed as the graph obtained from Cm and H := H +u joined by the path with two pendant vertices v and u. Applying Lemma 3.3, we have mG (α) ≤ mH (α) + 2. Noting that θ(H) = θ(G) − 1, p(H) = p(G) = 0 and mH (α) ≤ 2θ(H) − 1 (using the induction hypothesis), we have mG (α) ≤ mH (α) + 2 ≤ 2θ(H) − 1 + 2 ≤ 2θ(G) − 1. Case 3 G has no pendant vertices and a pair of cycles of G have common edges. Let e = u v be a common edge of two cycles Ck and Cl . Let u (resp., v) be a major vertex of G which minimizes dG−e (u , u) (resp., dG−e (v , v)). It should be indicated that u = u if u itself is a major vertex. Let P be the path beginning at u, across u , v , and ending at v and let H = G −P , where P = P −u −v. Then G is the graph obtained from H and P by identifying two pendant vertices of P respectively with two vertices u, v of H. If H is a cycle, then G is a θ-type bicyclic graph and we are done by Lemma 3.4. Suppose
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that H is not a cycle. Then mH (α) ≤ 2θ(H) − 1 (using the induction hypothesis). By Lemma 3.4, mG (α) ≤ mH (α) + 2. Noting that θ(G) = θ(H) + 1, p(G) = p(H) = 0, we have mG (α) ≤ mH (α) + 2 ≤ 2θ(H) − 1 + 2 ≤ 2θ(G) − 1. The proof for the claim is completed. If G is a cycle, the example has proven mG (α) ≤ 2 = 2θ(G) + p(G). Therefore, mG (α) ≤ 2 = 2θ(G) + p(G) for all connected graph G. If mG (α) = 2θ(G)+p(G), then the claim just proved (that is mG (α) ≤ 2θ(G)+p(G)−1 if G is a connected graph that is not a cycle) indicates that G is a cycle. Since the Aα spectrum of a cycle Cn is {2α+2(1 −α) cos 2πj n : j = 0, 1, . . . , n −1}, we have mCn (α) = 2 if and only if α =
2 cos 2πj n −1+2 cos 2πj n
with 0 < j <
n 2.
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