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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.

Bicyclic Graphs with the Second-Maximum and Third-Maximum Degree Resistance Distance