Spectral radii and star-factors with large components

TL;DR

This paper establishes spectral radius bounds for connected graphs with high isolated toughness to guarantee the existence of specific star-factor subgraphs, and demonstrates the sharpness of these bounds through extremal graph constructions.

Contribution

It provides new spectral bounds related to star-factors in isolated tough graphs, extending previous results and introducing sharpness via extremal examples.

Findings

Lower bounds on adjacency spectral radius for star-factor existence

Lower bounds on signless Laplacian spectral radius for star-factors

Upper bounds on distance spectral radius for graphs with star-factors

Abstract

Let $G$ be a connected graph with $n$ vertices. The isolated toughness of $G$, denoted by $I(G)$, is defined by $I(G)=\min\left\{\frac{|S|}{i(G-S)}:S\subseteq V(G) \ \mbox{and} \ i(G-S)\geq2\right\}$ if $G$ is not complete, or $I(G)=+\infty$ if $G$ is complete. A graph $G$ is called isolated $r$-tough if $I(G)\geq r$. A spanning subgraph $H$ of $G$ is called a $\{K_{1,j}:m\leq j\leq2m\}$-factor of $G$ if every component of $H$ is isomorphic to an element of $\{K_{1,j}:m\leq j\leq2m\}$. Let $\rho(G)$, $q(G)$ and $\mu(G)$ denote the adjacency spectral radius, the signless Laplacian spectral radius and the distance spectral radius of $G$, respectively. Let $m$ and $b$ be two positive integers with $m\geq2$. In this paper, we first establish a lower bounds on the adjacency spectral radius of a connected isolated $\frac{mb-1}{b}$-tough graph $G$ to guarantees that $G$ contains a $\{K_{1,j}:m\leq j\leq2m\}$-factor. Second, we establish a lower bounds on the signless Laplacian spectral radius of a connected isolated $\frac{mb-1}{b}$-tough graph $G$ to ensures that $G$ contains a $\{K_{1,j}:m\leq j\leq2m\}$-factor. Finally, we create an upper bounds on the distance spectral radius of a connected isolated $\frac{mb-1}{b}$-tough graph $G$ with a $\{K_{1,j}:m\leq j\leq2m\}$-factor. Furthermore, we construct some extremal graphs to claim that all the bounds obtained in this paper are sharp.