The sum of the diagonal degree matrix and the adjacency matrix of the graph is called the signless Laplacian matrix,and the signless Laplacian matrix of the connected graph is a non-negative irreducible matrix,and its largest eigenvalue is called the signless Laplacian spectral radius.A graph that satisfies a difference of 1 between the number of edges and vertices is called a Bicyclic graph,and a graph that has a difference of 2 from the number of edges and vertices is called a Tricyclic graph.The spectral problem has always been ahot research problem in graph theory.In this paper,we deter-mine the structure of graphs with maximum signless Laplacian spectral radius in the class of Bicyclic graph and Tricyclic graph with no pendant,respectively.
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