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

TL;DR

This paper investigates the maximum signless Laplacian spectral radius in $C_{4}$-free $k$-cyclic graphs, identifying extremal structures and extending results to Laplacian spectra.

Contribution

It determines the extremal graphs with maximal signless Laplacian spectral radius among $C_{4}$-free $k$-cyclic graphs and specific unicyclic and bicyclic cases.

Findings

Identified extremal graphs with maximum spectral radius.

Extended results to Laplacian spectral radius.

Provided explicit structures for maximal spectral cases.

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.