Unicyclic graphs with extremal Kirchhoff index

Abstract

Resistance distance was introduced by Klein and Randic (J. Math. Chem. 12 (1993) 81-95). The Kirchhoff index K-f(G) of a graph G is the sum of resistance distances between all pairs of vertices. Let S-n(l) denote the graph obtained from cycle C-l by adding n - l pendant edges to a vertex of C-l. Let P-n(l) denote the graph obtained by identifying one endvertex of path Pn-l+1 with any vertex of Cl. In this paper, we show that among n-vertex unicyclic graphs, (i) if n < 8, C-n has minimal Kirchhoff index; if 8 <= n < 12, S-n(4) has minimal Kirchhoff index; if n = 12, both S-n(3) and S-n(4) have minimal Kirchhoff index; otherwise, S-n(3) has minimal Kirchhoff index; (ii) P-n(3) has maximal Kirchhoff index. Sharp bounds for Kirchhoff index of unicyclic graphs are also obtained.