The signless Laplacian spectral radius of graphs with forbidding linear forests

TL;DR

This paper investigates the maximum signless Laplacian spectral radius in graphs avoiding certain linear forests, establishing stability results and characterizing extremal graphs for specific forbidden subgraphs.

Contribution

It provides new stability results and characterizations of extremal graphs for the spectral Turán problem involving linear forests with at most two odd paths.

Findings

Established a stability result for $k\cdot P_3$.

Determined extremal graphs avoiding fixed linear forests.

Maximized signless Laplacian spectral radius under given constraints.

Abstract

Tur\'{a}n type extremal problem is how to maximize the number of edges over all graphs which do not contain fixed forbidden subgraphs. Similarly, spectral Tur\'{a}n type extremal problem is how to maximize (signless Laplacian) spectral radius over all graphs which do not contain fixed subgraphs. In this paper, we first present a stability result for $k\cdot P_3$ in terms of the number of edges and then determine all extremal graphs maximizing the signless Laplacian spectral radius over all graphs which do not contain a fixed linear forest with at most two odd paths or $k\cdot P_3$ as a subgraph, respectively.