A class of bicyclic graphs with weighted Kirchhoff index
[Objective]We aim to investigate the weighted Kirchhoff index formula for bicyclic graphs G with n+2 vertices and consists of two cycles with a single common vertex.[Methods]By means of comparative coefficient and enumeration,we initially categorize the distribution of any two vertices vi and vj in the bicyclic graph G,and then discuss the distribution of e-vertices that are not saturated by an n-matching M in S(G)-vi-vj.[Results]Following a comprehensive classification discussion of three distinct distributional cases of vertices vi and vj,we establish that the weighted Kirchhoff index formula for bicyclic graphs,as derived in this paper,remains valid in each case.Subsequently,this result generalizes corresponding results of Li,Li,and Yan.on unicyclic graphs to bicyclic graphs with two cycles and only one common vertex.[Conclusion]This study demonstrates that the weighted Kirchhoff index formula for a bicyclic graph consisting of two cycles with a single common vertex can be computationally obtained and expressed for the sum of matching weights of its associated vertex-edge weighted subdivision graph S(G)ω*and its corresponding subgraphs.
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维普
