On the signless Laplacian spectral radius of $C_{4}$-free $k$-cyclic graphs

Abstract

A k-cyclic graph is a connected graph of order n and size n + k − 1. In this paper, we determine the maximal signless Laplacian spectral radius and the corresponding extremal graph among all C 4 -free k-cyclic graphs of order n. Furthermore, we determine the first three unicyclic, and bicyclic, C 4 -free graphs whose spectral radius of the signless Laplacian is maximal. Similar results are obtained for the (combinatorial) Laplacian.