Connected graphs with a large dissociation numberattaining the minimum spectral radius

Abstract

A dissociation set in a graph is a subset of vertices that induces a subgraph of maximum degree at most one, which is a natural generalization of the notion of an independent set. The dissociation number of a graph is defined as the maximum cardinality of a dissociation set. This paper studies the minimum spectral radius of connected graphs with a given order $n$ and a given dissociation number $ψ$. For $ψ=n-k$ with $k\ge 4$ fixed and $n$ sufficiently large, we establish both upper and lower bounds for this minimum spectral radius and prove the extremal graphs must belong to a specific graph class.