The Laplacian polynomial and Kirchhoff index of graphs based on R-graphs

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• 作者：Qun Liu^a ; ^{liuqun09@yeah.net" class="auth_mail" title="E-mail the corresponding author}Author Vitae ; Jia-Bao Liu^b ; ^{liujiabaoad@163.com" class="auth_mail" title="E-mail the corresponding author}Author Vitae ; Jinde Cao^c ; ^d ; ^{jdcao@seu.edu.cn" class="auth_mail" title="E-mail the corresponding author}Author Vitae
• 关键词：Kirchhoff index ; Resistance distance ; Corona ; Edge corona ; Laplacian polynomial ; Schur complement
• 刊名：Neurocomputing
• 出版年：2016
• 出版时间：12 February 2016
• 年：2016
• 卷：177
• 期：Complete
• 页码：441-446
• 全文大小：359 K

文摘

Let R(G)$R\left(G\right)$ be the graph obtained from G by adding a new vertex corresponding to each edge of G   and by joining each new vertex to the end vertices of the corresponding edge. Let I(G)$I\left(G\right)$ be the set of newly added vertices. The R-vertex corona of G₁ and G₂, denoted by 03f3388ef7b2d60a9db428a586200a7" title="Click to view the MathML source">G₁⊙G₂$G_1\odot G_2$, is the graph obtained from vertex disjoint R(G₁)$R\left(G_1\right)$ and |V(G₁)|$\left|V\right(G_1\left)\right|$ copies of G₂ by joining the i  th vertex of V(G₁)$V\left(G_1\right)$ to every vertex in the ith copy of G₂. The R-edge corona of G₁ and G₂, denoted by G₁⊖G₂$G_1\ominus G_2$, is the graph obtained from vertex disjoint R(G₁)$R\left(G_1\right)$ and f45b24a85d9479480" title="Click to view the MathML source">|I(G₁)|$\left|I\right(G_1\left)\right|$ copies of G₂ by joining the i  th vertex of I(G₁)$I\left(G_1\right)$ to every vertex in the ith copy of G₂. Liu et al. gave formulae for the Laplacian polynomial and Kirchhoff index of RT(G)$\mathit{RT}\left(G\right)$ in [19]. In this paper, we give the Laplacian polynomials of 03f3388ef7b2d60a9db428a586200a7" title="Click to view the MathML source">G₁⊙G₂$G_1\odot G_2$ and G₁⊖G₂$G_1\ominus G_2$ for a regular graph G₁ and an arbitrary graph G₂; on the other hand, we derive formulae and lower bounds of Kirchhoff index of these graphs and generalize the existing results.