Graph invariants, based on the distances between the vertices of a graph, are
widely used in theoretical chemistry. Recently, Gutman, Feng and Yu
(Transactions on Combinatorics, 01 (2012) 27- 40) introduced the degree
resistance distance of a graph G, which is defined as DR(G) = ?{u,v}?V(G)[dG(u)+dG(v)]RG(u,v),
where dG(u) is the degree of vertex u of the graph G, and RG(u, v)
denotes the resistance distance between the vertices u and v of the graph G.
Further, they characterized n-vertex unicyclic graphs having minimum
and second minimum degree resistance distance. In this paper, we characterize
n-vertex bicyclic graphs having maximum degree resistance distance.
Let H be a class of given graphs. A graph G is said to be H-free if G contains no induced copies of H for any H∈H. In this article, we characterize all connected subgraph pairs {R,S} guranteeing the edge-connectivity of a connected {R,S}-free graph to have the same minimum degree. Our result is a supplement of Wang et al. Furthermore, we obtain a relationship of forbidden sets when those general parameters have the recurrence relation.
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