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Let B(n, r) be the set of all bicyclic graphs with n vertices and r cut edges. In this paper we determine the unique graph with maximal adjacency spectral radius or signless Laplacian spectral radius among all graphs in B(n, r).
a r t i c l e i n f o a b s t r a c t When the underlying topology of an interconnection network is modeled by a connected graph G, the connectivity of G is an important measurement for reliability and fault tolerance of the network. For a given integer h 0, a subset F of edges in a connected graph G is an h-extra edge-cut if G − F is disconnected and every… (More)
A parity subgraph of a graph is a spanning subgraph such that the degrees of all vertices have the same parity in both the subgraph and the original graph. Let G be a cyclically 6-edge-connected cubic graph. Steffen (Intersecting 1-factors and nowhere-zero 5-flows 1306.5645, 2013) proved that G has a nowhere-zero 5-flow if G has two perfect matchings with… (More)
A subset F of edges in a connected graph G is a h-extra edge-cut if G − F is disconnected and every component has more than h vertices. The h-extra edge-connectivity λ (h) (G) of G is defined as the minimum cardinality over all h-extra edge-cuts of G.