The minimal degree Kirchhoff index of bicyclic graphs

Abstract

The degree Kirchhoff index of graph $ G $ is defined as $ Kf^{*}(G) = \sum\limits_{{u, v}\subseteq V(G)}d(u)d(v)r_{G}(u, v) $, where $ d(u) $ is the degree of vertex $ u $ and $ r_{G}(u, v) $ is the resistance distance between the vertices $ u $ and $ v $. In this paper, we characterize bicyclic graphs with exactly two cycles having the minimum degree Kirchhoff index of order $ n\geq5 $. Moreover, we obtain the minimum degree Kirchhoff index on bicyclic graphs of order $ n\geq4 $ with exactly three cycles, and all bicyclic graphs of order $ n\geq4 $ where the minimum degree Kirchhoff index has been obtained.

Keywords

• bicyclic graph
• degree kirchhoff index
• resistance distance
• $ \theta $-graph