On the spectral radii and the signless Laplacian spectral radii of c-cyclic graphs with fixed maximum degree

Abstract

If G is a connected undirected simple graph on n vertices and n + c - 1 edges, then G is called a c-cyclic graph. Specially, G is called a tricyclic graph if c = 3 . Let Δ ( G ) be the maximum degree of G. In this paper, we determine the structural characterizations of the c-cyclic graphs, which have the maximum spectral radii (resp. signless Laplacian spectral radii) in the class of c-cyclic graphs on n vertices with fixed maximum degree Δ ⩾ n + c + 1 2 . Moreover, we prove that the spectral radius of a tricyclic graph G strictly increases with its maximum degree when Δ ( G ) ⩾ 1 + 6 + 2 n 3 2 , and identify the first six largest spectral radii and the corresponding graphs in the class of tricyclic graphs on n vertices.