Maxima of the $Q$-index of non-bipartite graphs: forbidden short odd cycles

TL;DR

This paper determines the maximum signless Laplacian spectral radius (Q-index) for non-bipartite graphs avoiding short odd cycles, characterizing extremal graphs for fixed order or size.

Contribution

It uniquely identifies the extremal graphs and maximum Q-index for non-bipartite graphs with forbidden short odd cycles, considering fixed order or size.

Findings

Maximum Q-index determined for fixed order and forbidden short odd cycles.

Unique extremal graphs characterized for each case.

Results extend spectral graph theory with cycle restrictions.

Abstract

Let $G$ be a non-bipartite graph which does not contain any odd cycle of length at most $2k+1$. In this paper, we determine the maximum $Q$-index of $G$ if its order is fixed, and the corresponding extremal graph is uniquely characterized. Moreover, if the size of $G$ is given, the maximum $Q$-index of $G$ and the unique extremal graph are also proved.