Kirchhoff index in line, subdivision and total graphs of a regular graph

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Abstract

Let G be a connected regular graph and l(G), s(G), t(G) the line, subdivision, total graphs of G, respectively. In this paper, we derive formulae and lower bounds of the Kirchhoff index of l(G), s(G) and t(G), respectively. In particular, we give special formulae for the Kirchhoff index of l(G), s(G) and t(G), where G is a complete graph Kn, a regular complete bipartite graph Kn,n and a cycle Cn.

Keywords

Kirchhoff index
Line graph
Subdivision graph
Total graph

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This research was partially supported by the National Natural Science Foundation of China (No. 10971086 and No. 11126185) and the Fundamental Research Funds for the Central Universities (No. lzujbky-2011-46).