Abstract
Let G = V , E be a connected graph. The resistance distance between two vertices u and v in G , denoted by R G u , v , is the effective resistance between them if each edge of G is assumed to be a unit resistor. The degree resistance distance of G is defined as D R G = ∑ u , v ⊆ V G d G u + d G v R G u , v , where d G u is the degree of a vertex u in G and R G u , v is the resistance distance between u and v in G . A bicyclic graph is a connected graph G = V , E with E = V + 1 . This paper completely characterizes the graphs with the second-maximum and third-maximum degree resistance distance among all bicyclic graphs with n ≥ 6 vertices.
Highlights
All graphs considered in this paper are simple and undirected
Let G be a connected graph with a pendant vertex v with its unique neighbor w. en, Kfv(G) Kfw(G − v) + n − 1
Let G be a bicyclic graph of order n, vbe a pendant vertex of G, and w be its neighbor. en, DR(G) DR(G − v) + Dw(G − v) +2Kfw(G − v) + 3n
Summary
All graphs considered in this paper are simple and undirected. Let G (V, E) be a graph with n vertices and m edges. Let NG(v) be the set of vertices adjacent to v in G. e degree of v in G, denoted by dG(v), is equal to |NG(v)|. E distance between two vertices u and v of G, denoted by dG(u, v) or d(u, v), is the length of a shortest path connecting u and v in G. Klein and Randic [6] proved that RG(u, v) ≤ dG(u, v), with equality if and only if there is a unique path connecting u and v in G In recent years, this new type of distance between vertices in a graph has attracted prominent attention in mathematics and chemistry [6,7,8,9,10,11]. The bicyclic graphs with maximum and minimum DR-values were determined in [23, 24], respectively
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