On extremal bipartite bicyclic graphs

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• 作者：Jing Huang^a ; ^{1042833291@qq.com" class="auth_mail" title="E-mail the corresponding author} ; Shuchao Li^a ; ^{lscmath@mail.ccnu.edu.cn" class="auth_mail" title="E-mail the corresponding author} ; Qin Zhao^b
• 关键词：Estrada index ; Kirchhoff index ; Bipartite bicyclic graph ; Complement graph ; Closed walk
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2016
• 出版时间：15 April 2016
• 年：2016
• 卷：436
• 期：2
• 页码：1242-1255
• 全文大小：487 K

文摘

Let ${\mathcal{B}}_n^{+}$ be the set of all connected bipartite bicyclic graphs with n vertices. The Estrada index of a graph G   is defined as $\mathit{EE}(G)={\sum }_{i=1}^n{e}^{{\lambda }_i}$, where λ₁,λ₂,…,λ_n${\lambda }_1,{\lambda }_2,\dots ,{\lambda }_n$ are the eigenvalues of the adjacency matrix of G, and the Kirchhoff index of a graph G   is defined as Kf(G)=∑_i<jr_ij$\mathit{Kf}(G)={\sum }_{i<j}{r}_{ij}$, where r_ij$r_{ij}$ is the resistance distance between vertices v_i$v_i$ and v_j$v_j$ in G. The complement of G   is denoted by $\bar{G}$. In this paper, sharp upper bound on EE(G)$\mathit{EE}(G)$ (resp. $\mathit{Kf}(\bar{G})$) of graph G   in ${\mathcal{B}}_n^{+}$ is established. The corresponding extremal graphs are determined, respectively. Furthermore, by means of some newly created inequalities, the graph G   in ${\mathcal{B}}_n^{+}$ with the second maximal EE(G)$\mathit{EE}(G)$ (resp. $\mathit{Kf}(\bar{G})$) is identified as well. It is interesting to see that the first two bicyclic graphs in ${\mathcal{B}}_n^{+}$ according to these two orderings are mainly coincident.