On extremal bipartite bicyclic graphs

Accepted Manuscript On extremal bipartite bicyclic graphs

Jing Huang, Shuchao Li, Qin Zhao

PII: DOI: Reference:

S0022-247X(15)01185-3 http://dx.doi.org/10.1016/j.jmaa.2015.12.052 YJMAA 20081

To appear in:

Journal of Mathematical Analysis and Applications

Received date:

4 October 2014

Please cite this article in press as: J. Huang et al., On extremal bipartite bicyclic graphs, J. Math. Anal. Appl. (2016), http://dx.doi.org/10.1016/j.jmaa.2015.12.052

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On extremal bipartite bicyclic graphs∗ Jing Huanga , Shuchao Lia,† , Qin Zhaob a

Faculty of Mathematics and Statistics, Central China Normal University, Wuhan 430079, P.R. China b

Faculty of Mathematics and Statistics, Hubei University, Wuhan 430062, P.R. China

Abstract: Let Bn+ be the set of all connected bipartite bicyclic graphs with n vertices. The Estrada index of a n graph G is defined as EE(G) = i=1 eλi , where λ1 , λ2 , . . . , λn are the eigenvalues of the adjacency matrix of G,  and the Kirchhoff index of a graph G is defined as Kf (G) = i


rij = n

i
n−1  i=1

1 , μi

(1.1)

∗ Financially supported by the National Natural Science Foundation of China (Grant Nos. 11271149, 11371062) and the Program for New Century Excellent Talents in University (Grant No. NCET-13-0817). † Corresponding author. Email addresses: [email protected] (J. Huang), [email protected] (S.C. Li)

1

where μ1  μ2  · · ·  μn−1 > μn = 0 with n  2 are the eigenvalues of L(G). For more information on the Kirchhoff index, the readers are referred to recent papers [7, 9, 11, 19, 22] and references therein. n The Estrsds index of a graph G is defined as EE(G) = i=1 eλi , where λ1  λ2  · · ·  λn are the eigenvalues of A(G), called the eigenvalues of G. It was first proposed as a measure of the degree of folding of a protein [16]. It was found multiple applications of Estrsds index in a large variety of problems, including those in biochemistry and in complex networks, one may be referred to [17, 18, 20, 21] for details. Various properties, especially bounds for the Estrada index have been established in [8, 10, 13, 14, 15]. For a simple graph G, it is hard to recognize the relationship between EE(G) and Kf (G) (G the complement of G) from their definitions. However, on the one hand, we know that EE(G) is closely related to the numbers of closed walks in G; on the other hand, Deng and Chen [11] showed that Kf (G) is closely related to the numbers of closed walk in S(G) (the subdivision of G) if G is a bipartite graph. Furthermore, Deng and Chen [11] identified the first two and the last two trees according to the ordering of Kf (T ) by comparing the numbers of closed walks in S(T ), which coincide with the trees according to the ordering of EE(T ) for the first (resp. last) two graphs; see [8, 10, 25]. Recently, Deng and Chen [12] characterized the extremal connected bipartite unicyclic graphs with respect to both the Estrada index of themselves and the Kirchhoff index of their complements. They found that the first two graphs and the last one according to these two orderings are coincident. As a direct continuance of [11, 12], it is interesting for us to consider extremal bipartite bicyclic graphs with respect to the above two indices at the same time. A bicyclic graph G = (VG , EG ) is a connected simple graph which satisfies |EG | = |VG | + 1. There are two basic bicyclic graphs: ∞-graph and θ-graph. More concisely, an ∞-graph, denoted by ∞(p, q, l), is obtained from two vertex-disjoint cycles Cp and Cq by connecting one vertex of Cp and one of Cq with a path Pl of length l − 1( in the case of l = 1, identifying the above two vertices); and a θ-graph, denoted by θ(p, q, l), is an union of three internally disjoint paths Pp+1 , Pq+1 , Pl+1 of length p, q, l respectively with common end vertices, where p, q, l  1 and at most one of them is 1. Observe that any bicyclic graph G is obtained from an ∞-graph or a θ-graph by attaching trees to some of its vertices. The rest of the paper is arranged as follows. In Section 2, some preliminary results are given, which include three graph operations and their properties. In Section 3, we establish the sharp upper bound on EE(G) (resp. Kf (G)) of graph G in Bn+ . The corresponding extremal graphs are determined, respectively. In Section 4, the graph G in Bn+ with the second maximal EE(G) (resp. Kf (G)) is identified as well. In the Appendix, some new inequalities are established, which are used to prove our main results. 2. Preliminaries In this section, we give some necessary results which will be used to prove our main results. Let G be a simple graph with n vertices and m edges and put G to be the complement of G. In all what follows in this paper, we shall denote by Φ(B) = det(xI − B) the characteristic polynomial of the square matrix B. In particular, if B = A(G), we write Φ(A(G)) by χ(G; x) and call χ(G; x) the character-

2

istic polynomial of G; if B = L(G), we write Φ(L(G)) by σ(G; x) and call σ(G; x) the Laplacian characteristic polynomial of G. By [1], we have (n − x)σ(G; x) = (−1)n−1 xσ(G; n − x).

(2.1)

Further, if G is a bipartite graph, then χ(S(G); x) = xm−n σ(G; x2 ),

(2.2)

where S(G) is the subdivision graph of G, that is, the graph obtained from G by inserting a new vertex in each edge of G; see [30]. n Let Mk (G) be the k-th spectral moment of a graph G of order n, i.e., Mk (G) = i=1 λki (G). It is well known [6] that Mk (G) is equal to the numbers of closed walks of length k in G. In view of (1.1), (2.1) and (2.2), it is straightforward to deduce the following lemma. Lemma 2.1. Let G be a bipartite graph with n (n ≥ 2) vertices and m edges. Then Kf (G) =

n−m 1  + M2k (S(G))n−k − 1. 2 2 k0

Lemma 2.1 has been proved in both [11] and [12] by different methods. For any graph G ∈ Bn+ (the set of all bipartite bicyclic graphs with n vertices) and noting that, since G is a bipartite graph with |EG | = |VG | + 1, from the definition of the Estrada index and Lemma 2.1 for the Kirchhoff index, it follows that EE(G)

=

+∞  M2k (G)

(2k)!

k=0

Kf (G)

=

,

1  M2k (S(G)) 3 − . 2 nk 2

(2.3) (2.4)

k0

Remark 1. In view of (2.3) and (2.4), EE(G) and Kf (G) have certain similarity. That is, for G1 , G2 ∈ Bn+ , if M2k (G1 )  M2k (G2 ) for all nonnegative integers k, then EE(G1 )  EE(G2 ). Furthermore, if there exists one positive integer k0 such that M2k0 (G1 ) > M2k0 (G2 ), then EE(G1 ) > EE(G2 ); and the same thing is true for Kf (G1 ) and Kf (G2 ) by considering the number of closed walks in S(G) instead of that of itself. For simplicity, now we introduce some notation which will be used later. We will use G − x or G − xy to denote the graph that arises from G by deleting the vertex x ∈ V (G) or the edge xy ∈ E(G) (this notation is extended if more than one vertex or edge is deleted). Similarly, G + xy is a graph that arises from G by adding an edge xy ∈ E(G), where x, y ∈ V (G). For u, v ∈ VG , let Wk (G; u, v) denote the set of (u, v)-walks of length k in G, and put Wk (G; u, v) = |Wk (G; u, v)|. If u = v, we write Wk (G; u), Wk (G; u) instead of Wk (G; u, u), Wk (G; u, u) respectively. Note that Wk (G; u, v) = Wk (G; v, u) for all positive integer k; see [6]. Further, for two distinct vertices u, v ∈ VG , let Wk (G; u, [v]) be the set of (u, u)-walks of length k in G containing v, and let Wk (G; u, [v]) = |Wk (G; u, [v])|. Let G1 and G2 be two graphs with u1 , v1 ∈ VG1 and u2 , v2 ∈ VG2 . If Wk (G1 ; u1 , v1 )  Wk (G2 ; u2 , v2 ) for all positive integer k, then we write (G1 ; u1 , v1 )  (G2 ; u2 , v2 ) and there is at least one positive integer k0 such that Wk0 (G1 ; u1 , v1 ) < Wk0 (G2 ; u2 , v2 ), then we write (G1 ; u1 , v1 ) ≺ (G2 ; u2 , v2 ). Further writing (G; u) instead of (G; u, u). The coalescence G(u) · H(v) of rooted graphs G and H is the graph obtained from G and H by identifying the root u of G with the root v of H. Further on, we need the following lemmas. Lemma 2.2 ([29]). Given two bipartite graphs G1 and and W2k (G1 ; w)  W2k (G2 ; u) for any positive integer where G ∼ = G2 (u) · G3 (a) (see = G1 (w) · G3 (a) and G ∼ some positive integer k, then there must exist a positive

G2 with w ∈ VG1 and u ∈ VG2 . If M2k (G1 )  M2k (G2 ) k, then M2k (G)  M2k (G ) for any positive integer k, Fig. 1). Furthermore, if W2k (G1 ; w) > W2k (G2 ; u) for integer k0 such that M2k0 (G) > M2k0 (G ). 3

G1

• w(a)

G3

G2

G∼ = G1 (w) · G3 (a)

• u(a)

G3

G ∼ = G2 (u) · G3 (a)

Figure 1: Graphs G and G considered in Lemma 2.2.

/ EH for i = 1, 2, . . . , r, Lemma 2.3 ([13]). Let H be a graph with u, v ∈ VH . Suppose that wi ∈ VH , and uwi , vwi ∈ where r is a positive integer. Let Eu = {uw1 , uw2 , . . . , uwr } and Ev = {vw1 , vw2 , . . . , vwr }. Let Hu = H + Eu and Hv = H + Ev . If (H; u) ≺ (H; v) and (H; wi , u)  (H; wi , v) for 1  i  r, then Mk (Hu )  Mk (Hv ) for all positive integer k and it is strict for some positive integer k0 . Let A, B, C be three connected graphs, and each of which has at least two vertices. Let u, v be two different vertices of C, u a vertex of A (v  a vertex of B), then we define H = A(u ) · C(u);

G = H(v) · B(v  );

G = H(u) · B(v  ),

where the vertex u in H denotes the corresponding vertex of the coalescence of u in A and u in C (see Fig. 2 for an example). Then we have the following results. B A

C •u (u)

A

C  •u (u)

B •v  (v)

•u(v  ) C G

G

H

A

Figure 2: Graphs H, G and G considered in Lemma 2.4.

Lemma 2.4 ([12]). For the notation as above. Suppose that there exists an automorphism ϕ of C such that ϕ(u) = v and dH (u) > dH (v), then (i) (H; u) (H; v), that is, Wk (H; u)  Wk (H; v) for all positive integer k and it is strict for some positive integer k0 . (ii) Mk (G )  Mk (G) for all positive integer k and it is strict for some positive integer k0 . Remark 2. Note that, if A, B are trees and C is a path, then G is just the graph obtained from G by a proper generalized tree shift (GTS) defined in [5]. Since in this case, C naturally satisfies the condition of Lemma 2.4, thus a direct consequence of Lemma 2.4 is that the proper GTS increases (nondecreasing) the number of closed walks of length k, which was proved in [2] and [5] by using different methods. In the following, we call G the extended shift (ES) of G at u, v provided C satisfies the condition of Lemma 2.4 and write G = ES(G; u, v). 3. Bipartite bicyclic graphs with the maximum indices In this section, we determine extremal graphs in Bn+ with the maximum values of Kf (G) and EE(G) respectively. Let G = ES(G), then S(G ) = ES(S(G)). Note that if G is a bipartite graph. Hence, by Lemma 2.4 we have EE(G ) > EE(G),

Kf (G ) > Kf (G).

at the same time. So in the following we shall exploit the same effect of the extended shift on the two indices by giving parallel results without proofs.

4

...

...

v1

v4

. . .

v1

. . .

v3

v3 ...

v2

v4

v2

...

...

...

G

G

Figure 3: Graphs G and G considered in Lemma 3.2.

Note that every connected bicyclic graph can be viewed as a graph which is obtained by attaching trees to some vertices of ∞(p, q, l) or θ(p, q, l), and each tree can be transformed into a star by a sequence of applications of Lemma 2.2 or 2.4. Let Bs1 (p, q, l) (resp. Bs2 (p, q, l)) denote the subset of Bn+ in which the attaching trees to the ∞(p, q, l) (resp. θ(p, q, l)) are all stars. Note that any graph in Bs1 (p, q, l) can be transformed to a graph in Bs1 (p, q, 1) by a sequence of applications of Lemma 2.2 or 2.4. Then we immediately deduce the following result. Lemma 3.1. If G is the graph  with maximum Estrada index EE(G) or maximum Kirchhoff index Kf (G) among Bn+ . Then G ∈ Bs1 (p, q, 1) Bs2 (p, q, l). In the following, we first show that any Estrada maximum graph G in Bs1 (p, q, 1) will have a smaller Estrada index than a graph G in Bs2 (p − 1, q − 1, 1), and the same thing is true for Kf (G) and Kf (G ). And then investigate some properties of Estrada index EE(G) and Kirchhoff index Kf (G) among Bs2 (p, q, l). Finally, with the help of these properties, we determine the extremal graphs in Bs2 (p, q, l) with respect to both the Estrada index of themselves and the Kirchhoff index of their complements and find that the maximum bicyclic graphs in Bs2 (p, q, l) according to these two orderings are coincident. Lemma 3.2. For any G in Bs1 (p, q, 1), their exists a graph G in Bs2 (p − 1, q − 1, 1) such that EE(G) < EE(G ) and Kf (G) < Kf (G ). Proof. Let v1 , v2 , v3 , v4 be the vertices of G such that dG (v2 )  dG (v3 ) and v1 v2 , v1 v3 , v3 v4 ∈ EG ; see Fig. 3. Let H = G − {v3 v4 } and G = H + {v2 v4 }. On the one hand, we construct a mapping f from Wk (H; v3 ) to Wk (H; v2 ) as follows. If W ∈ Wk (H; v3 ) doesn’t reach the vertex v1 , then let f (W ) be the walk obtained from W by replacing every v3 and it’s pendent neighbors in W by v2 and it’s corresponding pendent neighbors, respectively. Otherwise, we may uniquely decompose W into three sections W1 , W2 , W3 , where W1 is the shortest (v3 , v1 )-section at the beginning of W, W2 is the longest (v1 , v1 )-section in the middle of W , and W3 is the remaining (v1 , v3 )-section at the end of W .In this case, we let f (W ) be the walk obtained from W by replacing every v3 and it’s pendent neighbors in W1 W3 by v2 and it’s corresponding pendent neighbors, respectively, and let W2 remain unchanged. Obviously, f (W ) ∈ Wk (H; v2 ) and f is an injection, i.e., Mk (H; v3 )  Mk (H; v2 ). Since dH (v3 ) = dG (v3 ) − 1  dG (v2 ) − 1 < dG (v2 ) = dH (v2 ), we have M2 (H; v3 ) < M2 (H; v2 ). This implies that (H; v3 ) ≺ (H; v2 ). On the other hand, we construct a mapping g from Wk (H; v4 , v3 ) to Wk (H; v4 , v2 ) as follows. For W  ∈ Wk (H; v4 , v3 ), we may uniquely decompose W  into two sections W1 , W2 , where W1 is the longest (v4 , v1 )-section of W  , and W2 is the remaining (v1 , v3 )-section of W  . In this case, we let g(W  ) be the walk obtained from W  by replacing every v3 and it’s pendent neighbors in W2 by v2 and it’s corresponding pendent neighbors, respectively, and W1 remain unchanged. Obviously, g(W  ) ∈ Wk (H; v4 , v2 ) and g is an injection, i.e., Mk (H; v4 , v3 )  Mk (H; v4 , v2 ) for all positive integer k. Note that G = H + {v3 v4 } and G = H + {v2 v4 }. Hence, by Lemma 2.3 and Eq.(2.3), we have EE(G) < EE(G ). Let S(G) and S(G ) be the subdivision of G and G , respectively, and let H  = S(G) − {v3 v3,4 }, where v3,4 denotes the subdividing vertex of edge v3 v4 . Then in a similar way, we have (H  ; v3 ) ≺ (H  ; v2 ) and (H  ; v3,4 , v3 )  (H  ; v3,4 , v2 ). Note that S(G) = H  + {v3 v3,4 } and S(G ) = H  + {v2 v3,4 }. Hence, by Lemma 2.3 and Eq.(2.4), we have Kf (G) < Kf (G ). Moving all the pendant edges from v3 to v1 in graph G yields a graph, say G . That is G = ES(G ). It is routine to check that G is in Bs2 (p − 1, q − 1, 1). Hence, our proof is completed.

5

Let G ∈ Bs2 (p, q, l), where Pp+1 = v1 v2 . . . vp vp+1 , Pl+1 = v1 u2 u3 . . . ul vp+1 and Pq+1 = v1 w2 w3 . . . wq vp+1 are the three internally disjoint paths with common end vertices v1 and vp+1 . Without loss of generality, assume that dG (v2 )  dG (u2 ). Put G = G − v2 v3 + u2 v3 . Lemma 3.3. Let G, G be defined as above. Then (i) EE(G) < EE(G ); (ii) Kf (G) < Kf (G ). Proof. (i) Let H = G − v2 v3 , then G = H + v2 v3 , G = H + u2 v3 . On the one hand, we construct a mapping f from Wk (H; v2 ) to Wk (H; u2 ) as follows. If W ∈ Wk (H; v2 ) doesn’t reach the vertex v1 , then let f (W ) be the walk obtained from W by replacing every v2 and it’s pendent neighbors in W by u2 and it’s corresponding pendent neighbors, respectively. Otherwise, we may uniquely decompose W into three sections W1 , W2 , W3 , where W1 is the shortest (v2 , v1 )-section at the beginning of W, W2 is the longest (v1 , v1 )-section in the middle of W , and W3 is the remaining (v1 , v2 )-section at the end of W . Inthis case, we let f (W ) be the walk obtained from W by replacing every v2 and it’s pendent neighbors in W1 W3 by u2 and it’s corresponding pendent neighbors, respectively, and let W2 remain unchanged. It is routine to check that f (W ) ∈ Wk (H; u2 ) and f is an injection, i.e., Mk (H; v2 )  Mk (H; u2 ). Since dH (v2 ) = dG (v2 ) − 1  dG (u2 ) − 1 < dG (u2 ) = dH (u2 ), we have M2 (H; v2 ) < M2 (H; u2 ). Thus (H; v2 ) ≺ (H; u2 ). On the other hand, we construct a mapping g from Wk (H; v3 , v2 ) to Wk (H; v3 , u2 ) as follows. For W  ∈ Wk (H; v3 , v2 ), we may uniquely decompose W  into two sections W1 , W2 , where W1 is the longest (v3 , v1 )section of W  , and W2 is the remaining (v1 , v2 )-section of W  . In this case, we let g(W  ) be the walk obtained from W  by replacing every v2 and it’s pendent neighbors in W2 by u2 and it’s corresponding pendent neighbors, respectively, and W1 remain unchanged. Obviously, g(W  ) ∈ Wk (H; v3 , u2 ) and g is an injection, i.e., Mk (H; v3 , v2 )  Mk (H; v3 , u2 ). Hence, by Lemma 2.3 and Eq.(2.3), we have EE(G) < EE(G ). (ii) Let S(G) and S(G ) be the subdivision of G and G , respectively, and let H  = S(G) − {v2 v2,3 }, where v2,3 denotes the subdividing vertex of edge v2 v3 . Then in a similar way, we have (H  ; v2 ) ≺ (H  ; u2 ) and (H  ; v2,3 , v2 )  (H  ; v2,3 , u2 ). Note that S(G) = H  + {v2 v2,3 } and S(G ) = H  + {u2 v2,3 }. Hence, by Lemma 2.3 and Eq.(2.4), we have Kf (G) < Kf (G ). Remark 3. Note that if move all the pendant edges from v2 to v1 of G in Lemma 3.3, we obtain a graph G . It is routine to check that G is in Bs2 (p − 1, q − 1, l + 1). Hence, for any G ∈ Bs2 (p, q, l), by repeatedly using Lemma 3.3, from G we can obtain a graph, say G0 , in Bs2 (2, 2, 2) such that its Estrada index EE(G0 ) (resp. Kirchhoff index Kf (G0 )) is as large as possible. So by Lemmas 3.1-3.3, we can find that if G is a graph with maximum Estrada index EE(G) (resp. maximum Kirchhoff index Kf (G)) among Bn+ , then G must be in Bs2 (2, 2, 2). In the following, we identify the graph with the maximum EE(G) (resp. Kf (G)) in Bs2 (2, 2, 2). n2 . v3

n3

. .

v1

v5 ...

n5

v4

...

n4

...

...

v2 . . .

n1

. v3

n3

. .

v1

v5

...

v2

v2

⇒

n2 + n4 + n5

n2 + n5

n2 + n5

...

. . .

n1

⇒ v3

v1

v5

. . .

⇒ n1 + n3

v4

v4

...

...

v2 v3

v1

v5

. . .

n1 + n3

v4

n4

n4

Figure 4: The extended shift of graph θ5 (n1 , n2 , n3 , n4 , n5 ). Let {v1 , v2 , v3 , v4 , v5 } be the set of all vertices of graph θ(2, 2, 2) with dθ(2,2,2) (v1 ) = dθ(2,2,2) (v3 ) = 3, dθ(2,2,2) (v2 ) = dθ(2,2,2) (v4 ) = dθ(2,2,2) (v5 ) = 2. Put θ5 (n1 , n2 , n3 , n4 , n5 ) to be the graph obtained from θ(2, 2, 2) by attaching ni pendant vertices to vi of θ(2, 2, 2), i = 1, 2, 3, 4, 5. It is routine to check that θ5 (n1 , n2 , n3 , n4 , n5 ) is in Bs2 (2, 2, 2). It is straightforward to check that graph θ5 (n1 + n3 , n2 + n4 + n5 , 0, 0, 0) can be obtained from θ5 (n1 , n2 , n3 , n4 , n5 ) by implementing extended shift at most three times, which may be viewed in Fig. 4. Lemma 3.4. Let G ∼ = θ5 (a + b, 0, 0, 0, 0), where a  0, b  1. Then = θ5 (a, b, 0, 0, 0), G ∼

6

b... v2 v1

v5

v3

v2 . . .

v5

v3

a

v4 G

v2 v1

. . .

v3

a+b

v5

v4 G

v1

. . .

a

v4 H

Figure 5: Graphs G, G and H considered in Lemma 3.4.

(i) EE(G) < EE(G ); (ii) Kf (G) < Kf (G ). Proof. (i) Let G, G and H be the graphs as depicted in Fig. 5. Then it is routine to check that G ∼ = H(v2 ) · K1,b (v2 )and G ∼ = H(v1 ) · K1,b (v1 ). For any W ∈ Wl (H; v2 , v3 ), replacing the last v3 by v1 in W yields a walk W  ∈ Wl (H; v2 , v1 ). Then for any positive integer l, we have Ml (H; v2 , v3 )  Ml (H; v2 , v1 ). In a similar way, M2l (S(H); v2 , v3 )  M2l (S(H); v2 , v1 ) for any positive integer l . Now for any k  1, noting that Mk (G; u, v) = Mk (G; v, u) for any u and v and by the symmetry of v2 , v4 and v5 in H, we have Mk (H; v1 )  = Mk (H; v2 ) =  =

Mk−1 (H; v1 , v2 ) + Mk−1 (H; v1 , v4 ) + Mk−1 (H; v1 , v5 ) 3Mk−1 (H; v1 , v2 ), Mk−1 (H; v2 , v1 ) + Mk−1 (H; v2 , v3 ) 2Mk−1 (H; v2 , v1 ) 2Mk−1 (H; v1 , v2 ).

So Mk (H; v2 )  Mk (H; v1 ) for any k  1. In particular, M2 (H; v2 ) = 2 < 3 + a = M2 (H; v1 ). Thus by Lemma 2.2 and Eq.(2.3), we have EE(G) < EE(G ). (ii) We first show by induction that for any non-negative integer k, M2k (S(H); v2 )  M2k (S(H); v1 ).

(3.1)

In fact, M0 (S(H); v2 ) = M0 (S(H); v1 ) = 1. Hence, (3.1) is true for k = 0. Suppose it is true for k. For the case k + 1, one has M2(k+1) (S(H); v1 )

= M2k (S(H); v1 , v2 ) + M2k (S(H); v1 , v4 ) + M2k (S(H); v1 , v5 ) + (a + 3)M2k (S(H); v1 ) = 3M2k (S(H); v1 , v2 ) + (3 + a)M2k (S(H); v1 )  3M2k (S(H); v1 , v2 ) + 3M2k (S(H); v1 )

whereas M2(k+1) (S(H); v2 )

= M2k (S(H); v2 , v1 ) + M2k (S(H); v2 , v3 ) + 2M2k (S(H); v2 )  2M2k (S(H); v2 , v1 ) + 2M2k (S(H); v2 ) = 2M2k (S(H); v1 , v2 ) + 2M2k (S(H); v2 ).

Hence, we have M2(k+1) (S(H); v2 )  M2(k+1) (S(H); v1 ) for all non-negative integer k. Note that M2 (S(H); v2 ) = 2 < 3 + a = M2 (S(H); v1 ) and S(G) = S(H)(v2 ) · S(K1,b )(v2 ), S(G ) = S(H)(v1 ) · S(K1,b )(v1 ). Hence, it follows from Lemma 2.2 and Eq.(2.4) that Kf (G) < Kf (G ). This completes the proof. Theorem 3.5. Let G be in Bn+ . Then 

n+1+

EE(G) Kf (G)

√

n2 −10n+61 2

 −

+e  n−4+e 4 3 2 2n − 17n + 49n − 50n − 4 .  (n − 1)(n − 2)(n − 5)

n+1+



√

n2 −10n+61 2

n+1−

+e

Each of the above equalities holds if and only if G ∼ = θ5 (n − 5, 0, 0, 0, 0). 7



√

n2 −10n+61 2

+e

−

n+1−

√

n2 −10n+61 2

,

Proof. For convenience, let G∗ := θ5 (n − 5, 0, 0, 0, 0). From Remark 3, Lemma 3.4, we have EE(G)  EE(G∗ ) and Kf (G)  Kf (G∗ ), each of the equalities holds if and only if G ∼ = G∗ . In what follows, we are to determine ∗ ∗ the values for both EE(G ) and Kf (G ). By direct calculation, we have χ(G∗ ; x) σ(G∗ ; x)

=

xn−4 [x4 − (n + 1)x2 + 3(n − 5)],

(3.2)

=

x(x − 1)n−6 (x − 2)2 [x3 − (n + 4)x2 + (5n − 2)x − 3n].

(3.3)

Let χ(G∗ ; x) = 0. In view of (3.2) it roots are x1 = x2 = · · · = xn−4 = 0 and   √ √ n + 1 + n2 − 10n + 61 n + 1 + n2 − 10n + 61 , xn−2 = − , xn−3 = 2 2   √ √ n + 1 − n2 − 10n + 61 n + 1 − n2 − 10n + 61 , xn = − . xn−1 = 2 2 Hence,  ∗

EE(G ) = n − 4 + e

n+1+



√

n2 −10n+61 2

+e

−

n+1+



√

n2 −10n+61 2

n+1−

+e



√

n2 −10n+61 2

+e

−

n+1−

√

n2 −10n+61 2

.

Now, let α1 , α2 , α3 denote the roots of φ(x) = x3 − (n + 4)x2 + (5n − 2)x − 3n = 0. In view of (3.3), and by Eqs.(1.1) and (2.1), we have   2 1 n−6 1 1 + + + + Kf (G∗ ) = n n − 1 n − 2 n − α1 n − α2 n − α3   n−6 2 φ (n) = n + + n−1 n−2 φ(n) 4 3 2 2n − 17n + 49n − 50n − 4 = . (n − 1)(n − 2)(n − 5) This completes the proof. The graph in Bn+ with the maximum Estrada index was first determined in [28], here we just use a different method, which is inspired by [12]. 4. Bipartite bicyclic graphs with the second maximum indices In this sectioin, we are to determine the graphs with the second maximum EE(G) or Kf (G) among Bn+ . We will show that, among Bn+ , the graph with the second maximum EE(G) has a little difference with the graph having the second maximum Kf (G). Lemma 4.1. Let G be in Bs2 (2, 2, 2) with uu1 ∈ EG , where uu1 is a non-cut edge. If there exists a pendant vertex, say u2 , being adjacent to u, then (i) (G; u2 ) ≺ (G; u1 ); (ii) (S(G); u2 ) ≺ (S(G); u1 ). Proof. (i) Let l be a positive integer, we construct a mapping f from Wl (G; u2 ) to Wl (G; u1 ). For W ∈ Wl (G; u2 ), let f (W ) be the walk obtained from W by replacing it’s first (resp. last) vertex u2 by u1 and leave the rest unchanged. Obviously, f (W ) ∈ Wl (G; u1 ) and f is an injection, i.e., Ml (G; u2 )  Ml (G; u1 ). Since dG (u1 ) > dG (u2 ) = 1, we have M2 (G; u2 ) < M2 (G; u1 ). This implies that (G; u2 ) ≺ (G; u1 ). (ii) Now consider the subdivision graph S(G). Suppose that v1 , v2 denote the subdividing vertices of the edges uu1 , uu2 in G, respectively. Let k be a positive integer, we construct a mapping g from Wk (S(G); u2 ) to Wk (S(G); u1 ). If W ∈ Wk (S(G); u2 ) doesn’t contain the vertex u, then let g(W ) be the walk obtained from W by replacing every u2 , v1 in W by u1 , v2 , respectively. Otherwise, we may uniquely decompose W into three sections 8

W1 , W2 , W3 , where W1 is the shortest (u2 , u)-section at the beginning of W , W2 is the longest (u, u)-section in the )-section at the end of W . In this case, we let g(W ) be the walk obtained middle of W , and W3 is the remain (u, u2 from W by replacing every u2 , v2 in W1 W3 by u1 , v1 , respectively, and let W2 remain unchanged. Obviously, g(W ) ∈ Wk (S(G); u1 ) and g is an injection, i.e., Mk (S(G); u2 )  Mk (S(G); u1 ). Since dS(G) (u1 ) > dS(G) (u2 ) = 1, we have M2 (S(G); u2 ) < M2 (S(G); u1 ). This implies that (S(G); u2 ) ≺ (S(G); u1 ). This completes the proof. v2 v3

w v v5 1

v4 H

v2 . . .

a

v3

v2

w

v5

v5

v3

v4 H1

v2

w a−1

v4 H2

. v3

w

v5

. .

v4 H3

Figure 6: The graphs considered in Lemma 4.2.

Lemma 4.2. Let G ∼ = θ5 (a + b − 1, 1, 0, 0, 0) with a  1, b  2. Then = θ5 (a, b, 0, 0, 0), G ∼ (i) EE(G) < EE(G ); (ii) Kf (G) < Kf (G ). Proof. Let H, H1 , H2 and H3 be four graphs with labelled vertices as depicted in Fig. 6. Then it is routine to check that G = H(v2 ) · K1,b−1 (v2 ), G = H(v1 ) · K1,b−1 (v1 ). On the one hand, Mk (H − v1 ; v2 ) = Mk (H1 ; v2 )  Mk (H2 ; v2 ), where the inequality follows from the fact that H1 is a subgraph of H2 . Then it follows from Lemma 2.4(i) that Mk (H2 ; v2 )  Mk (H2 ; v3 ) with strict inequality for at least one k. Since H2 is a subgraph of H3 , Mk (H2 ; v3 )  Mk (H3 ; v3 ) = Mk (H −v2 −w; v1 ) = Mk (H −v2 ; v1 ). Thus we have Mk (H − v1 ; v2 )  Mk (H − v2 ; v1 ) with strict inequality for at least one k. On the other hand, for each walk W ∈ Wk (H; v2 , [v1 ]), write it as W = W1 W2 , where W1 is the longest (v2 , v1 )section of W , and W2 is the remaining (v1 , v2 )-section of W . We construct a mapping f from Wk (H; v2 , [v1 ]) to Wk (H; v1 , [v2 ]) by f (W ) = W2 W1 . Obviously, f (W ) ∈ Wk (H; v1 , [v2 ]) and f is an injection and hence Mk (H; v2 , [v1 ])  Mk (H; v1 , [v2 ]). Note that Mk (H; v2 ) = Mk (H − v1 ; v2 ) + Mk (H; v2 , [v1 ]) and Mk (H; v1 ) = Mk (H − v2 ; v1 ) + Mk (H; v1 , [v2 ]). Hence, we have Mk (H; v2 )  Mk (H; v1 ). Since dH (v2 ) = 3 < 3 + a = dH (v1 ), we have M2 (H; v2 ) < M2 (H; v1 ). This implies that (H; v2 ) ≺ (H; v1 ). Then it follows from Lemma 2.2 and Eq.(2.3) that EE(G) < EE(G ). By a similar discussion as in the proof as above, we may also show that Kf (G) < Kf (G ), which is omitted here. This completes the proof. Since we hope to find the graph G among Bn+ with the second maximum EE(G) or the second maximum Kf (G), by Lemmas 3.2 and 3.3 G must be in Bs2 (2, 2, 2). For the graph G = θ5 (n1 , n2 , n3 , n4 , n5 ), it is easy to see that G is in Bs2 (2, 2, 2). If there are at least three non-zero ni ’s for θ5 (n1 , n2 , n3 , n4 , n5 ), then G can be changed into the graph G = θ5 (n1 + n3 , n2 + n4 + n5 , 0, 0, 0) by at most three extended shifts, which can be illustrated in Fig. 6. So EE(G) < EE(G ) and Kf (G) < Kf (G ). For positive integers a  b  1, let H := θ5 (b − 1, 0, b − 1, 0, 0)(v1 ) · K1,a−b+1 (v1 ). We have θ5 (a + 1, 0, b − 1, 0, 0) = H(v1 ) · K2 (v1 ),

θ5 (a, 0, b, 0, 0) = H(v3 ) · K2 (v3 ).

Hence, by Lemma 2.4 we have EE(θ5 (a, 0, b, 0, 0)) < EE(θ5 (a + 1, 0, b − 1, 0, 0) and Kf (θ5 (a, 0, b, 0, 0)) < Kf (θ5 (a + 1, 0, b − 1, 0, 0)). Combined with Lemmas 4.1 and 4.2, if G is a graph with the second maximum EE(G) or second maximum Kf (G) among Bn+ , then G must in {θ5 (n − 6, 0, 1, 0, 0), θ5 (n − 6, 1, 0, 0, 0), θ5 (0, n − 5, 0, 0, 0)}. For convenience, let G1 := θ5 (n−6, 0, 1, 0, 0), G2 := θ5 (n−6, 1, 0, 0, 0), G3 := θ5 (0, n−5, 0, 0, 0) in what follows. Theorem 4.3. Let G ∈ Bn+ \ {θ5 (n − 5, 0, 0, 0, 0)} with n  6. 9

∼ G1 ; (i) If 6  n  12, then Kf (G)  Kf (G1 ) with equality if and only if G = (ii) If n  13, then Kf (G)  Kf (G2 ) = Kf (G3 ) with equality if and only if G ∼ = G2 or G3 . Proof. By direct calculation, we have σ(G1 , x) = x(x − 1)n−7 (x − 2)2 [x4 − (n + 5)x3 + (7n − 4)x2 − 2(5n − 7)x + 3n]; σ(G2 , x) = σ(G3 , x) = x(x − 1)n−6 (x − 2)(x − 3)[x3 − (n + 3)x2 + (5n − 8)x − 2n]. Suppose x1 , x2 , x3 , x4 are the roots of f (x) = 0 with f (x) = x4 − (n + 5)x3 + (7n − 4)x2 − 2(5n − 7)x + 3n. Hence by Eq.(1.1) and Eq.(2.1), we have Kf (G2 ) = K(G3 ) and   2 1 n−7 1 1 1 + + + + + Kf (G1 ) = n n − 1 n − 2 n − x1 n − x2 n − x3 n − x4   2 f  (n) n−7 + + = n n−1 n−2 f (n)   n−7 2 n3 − n2 − 18n + 14 = n + + . n−1 n−2 2n3 − 14n2 + 17n In a similar way, we have  Kf (G2 ) = Kf (G3 ) = n

1 1 n2 − n − 8 n−6 + + + n−1 n−2 n−3 2n2 − 10n

 .

Hence, Kf (Gi ) − Kf (G1 ) =

n5 − 17n4 + 51n3 + 37n2 − 128n − 24 , (n − 1)(n − 2)(n − 3)(2n − 10)(2n2 − 14n + 17)

i = 2, 3.

Note that n5 − 17n4 + 51n3 + 37n2 − 128n − 24 < 0 when 6  n  12, and n5 − 17n4 + 51n3 + 37n2 − 128n − 24 > 0 when n  13. Whence, Theorem 4.3 holds. This completes the proof. Theorem 4.4. Let G ∈ Bn+ \ {θ5 (n − 5, 0, 0, 0, 0)} with n  6. (i) If 6  n  22, then EE(G)  EE(G1 ) with equality if and only if G ∼ = G1 ; (ii) If n  23, then EE(G)  EE(G2 ) with equality if and only if G ∼ = G2 . Proof. By direct calculation, we have χ(G1 , x) = χ(G2 , x) = χ(G3 , x) =  √

xn−4 [x4 − (n + 1)x2 + 4n − 21]; xn−6 [x6 − (n + 1)x4 + 4(n − 5)x2 − 2(n − 6)];

xn−4 [x4 − (n + 1)x2 + 4(n − 5)].  √ n+1+ n2 −14n+85 n+1+ n2 −14n+81 and λ (G ) = . By direct calculation, we have Then we have λ1 (G1 ) = 1 3 2 2 EE(G1 ) > EE(G2 ) = EE(G3 ) when n = 6, EE(G1 ) > EE(G2 ) > EE(G3 ) when 7  n  22 and EE(G2 ) > EE(G1 ) > EE(G3 ) when 23  n  120.  √

√

2

2

Now assume n  121. Since χ(G2 , n+1+ n2−14n+87 ) < 0, λ1 (G2 ) > n+1+ n2−14n+87 . Let u, v, w be the vertices of maximum degree in graph G1 , G2 and G3 , respectively. The graphs G1 − u and G3 − w has eigenvalues √ ±2 and 0 with multiplicity n − 3, the graph G2 − v has eigenvalues ± 2 ± 2 and 0 with multiplicity n − 5. By interlacing property of the eigenvalues of A(G1 −u) and A(G1 ) (see [6]), λi (G1 )  λi (G1 −u) for i = 2, 3, . . . , n−1. So EE(G1 ) =

n  i=1

e

λi (G1 )

>e

λ1 (G1 )

+

n−1 



e

λi (G1 −u)

i=2

10

n+1+

=e

√

n2 −14n+85 2

+ (n − 3) + e−2 .

Similarly, by the fact λi (G3 )  λi−1 (G3 − w) for i = 2, 3, . . . , n, EE(G3 ) =

n 

e

λi (G3 )

e

λ1 (G3 )

+

n−1 

i=1  n+1+



n2 −14n+85 2

n+1+

=

e

λi (G2 )

>e

λ1 (G2 )

n2 −14n+81 2

n−1 

+

i=1

EE(G1 )

n 

=

n+1+

√

n2 −14n+81 2

=e

+ (n − 3) + e−2 + e2 .

+ e2 for n  121 (see the Appendix). Thus we have EE(G1 ) >



e

λi (G2 −v)

e

λi (G1 −u)

n+1+

e

λi (G1 )

e

λ1 (G1 )

n−1 

+

√

√

n2 −14n+87 2

>e

i=2

i=1 

n+1+

√

Note that e >e EE(G3 ) for n  121. In a similar way, we have EE(G2 )

e

λi (G3 −w)

i=1

√

n 



 n+1+

+e

√ 2+ 2

√ +e

√ 2− 2

+ (n − 5);

√

=e

n2 −14n+85 2

+ e−2 + e2 + (n − 3).

i=1

√

√

n2 −14n+87 2

√

√ 2+ 2



√ 2− 2

n+1+

√

n2 −14n+85

2 Note that e +e +e >e +e−2 +e2 +2 for n  121 (see the Appendix). Then we have EE(G2 ) > EE(G1 ) for n  121. Thus, Theorem 4.4 holds. This completes the proof.

Appendix In this Appendix, we establish the following two inequalities: For n  121, one has   n+1+

√

n2 −14n+85 2

e and

 n+1+

√

√

e

n2 −14n+87 2

+e

√ 2+ 2

n+1+

√

>e

√ +e

n2 −14n+81 2



√ 2− 2

n+1+

+ e2

(A)

√

n2 −14n+85 2

>e

+ e−2 + e2 + 2.

(B)

Here we only show inequality (A). Similarly, we can also show inequality (B), which will be omitted here. Proof of inequality (A).

In fact, inequality (A) holds if and only if

 n+1+

√

e

n2 −14n+85 − 2

 n+1+



√

n2 −14n+81 2

>1+e

n+1+

2−

√

n2 −14n+81 2

.

Note that ex > x + 1 for x > 0. Hence,  n+1+

e  where f (n) =

√ n+1+ n2 −14n+85 2

−



√

n2 −14n+85 − 2

 n+1+

√ n+1+ n2 −14n+81 . 2

√

> f (n) + 1,

Hence, it suffices to show e2



f (n) >

n2 −14n+81 2

n+1+

e

√

n2 −14n+81 2

.

In fact, on the one hand,  f (n)

=

n+1+ √

=



 √ √ n2 − 14n + 85 n + 1 + n2 − 14n + 81 − 2 2

n2 −14n+85 2

√ n+1+ n2 −14n+85 2

− +

√

n2 −14n+81 2 √ n+1+ n2 −14n+81 2



11

(A1)



> 2 >



>

√

√

n+1+ n2 −14n+85 2

1 √ n+1+ n2 −14n+85 2

√

1

n2 −14n+85 2

e2 n+1+

e Note that

3 94 4



1 1 = 3 . n · 2n 2n 2 9

√

n2 −14n+81 2

1 · √ 2 n2 − 14n + 85

On the other hand, note that ex − x 2 > 0 if x > 10.8 and have 

√

+

n2 −14n+81 2

< 



√ n+1+ n2 −14n+81 2

> 10.8 if n  121. Hence, we

e2

e2 9 < 

94 . 3 √ 2 n n+1+ n2 −14n+81 4 2

3

· n 4 > 2e2 for n  121, then we have 1 2n

3 2

e2 > 9 . 3 4 4n

Thus, inequality (A1) holds. This completes the proof. Acknowledgements The authors would like to express their sincere gratitude to the referee for a very careful reading of this paper and for all the insightful comments, which led a number of improvements to this paper. References [1] N.L. Biggs, Algebraic Graph Theory, second ed., Cambridge Univ. Press, 1993. [2] B. Bollob´ as, M. Tyomkyn, Walks and paths in trees, J. Graph Theory 70 (2012) 54-66. [3] D. Bonchev, A.T. Balaban, X. Liu, D.J. Klein, Molecular cyclicity and centricity of polycyclic graphs. I: cyclicity based on resistance distances or reciprocal distances, Internat. J. Quantum Chem. 50 (1994) 1-20. [4] J.A. Bondy, U.S.R. Murty, Graph Theory, in: Graduate Texts in Mathematics, vol. 244, Springer, 2008. [5] P. Csikv´ ari, On a poset of trees, Combinatorica 30 (2010) 125-137. [6] D. Cvetkovi´c, M. Doob, H. Sachs, Spectra of Graphs-Theory and Application, Academic Press, New York, 1980. [7] K.C. Das, A.D. G¨ ung¨ or, A.S. C ¸ vik, On the Kirchhoff index and the resistance-distance energy of a graph, MATCH Commun. Math. Comput. Chem. 67 (2012) 541-556. [8] K.C. Das, S.G. Lee, On the Estrada index conjecture, Linear Algebra Appl. 431 (2009) 1351-1359. [9] K.C. Das, K.X. Xu, I. Gutman, Comparison between Kirchhoff index and the Laplacian-energy-like invariant, Linear Algebra Appl. 436 (2012) 3661-3671. [10] H.Y. Deng, A proof of a conjecture on the Estrada index, MATCH Commun. Math. Comput. Chem. 62 (2009) 599-606. [11] Q.Y. Deng, H.Y. Chen, On the Kirchhoff index of the complement of a bipartite graph, Linear Algebra Appl. 439 (2013) 167-173. [12] Q.Y. Deng, H.Y. Chen, On extremal bipartite unicyclic graphs, Linear Algebra Appl. 444 (2014), 89-99. [13] Z.B. Du, Z. Liu, On the Estrada and Laplacian Estrada indices of graphs, Linear Algebra Appl. 435 (2011) 2065-2076. [14] Z.B. Du, B. Zhou, The Estrada index of trees, Linear Algebra Appl. 435 (2011) 2462-2467. [15] Z.B. Du, B. Zhou, On the Estrada index of graphs with given number of cut edges, Electron. J. Linear Algebra 22 (2011) 586-592. [16] E. Estrada, Characterization of 3D molecular structure, Chem. Phys. Lett. 319 (2000) 713-718. [17] E. Estrada, Characterization of the folding degree of proteins, Bioinformatics 18 (2002) 697-704.

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