Connected graphs with a given dissociation number attaining the minimum spectral radius

TL;DR

This paper characterizes connected graphs with specified dissociation numbers that minimize spectral radius, identifying their structure and proving they are trees under certain conditions.

Contribution

It provides a complete characterization of such graphs for specific dissociation numbers and establishes their tree structure when the dissociation number exceeds a threshold.

Findings

Graphs with given dissociation numbers are characterized for minimal spectral radius.

Identifies the structure of these graphs for specific dissociation numbers.

Proves these graphs are trees when the dissociation number exceeds rac{2n}{3}.

Abstract

A dissociation set of a graph is a set of vertices which induces a subgraph with maximum degree less than or equal to one. The dissociation number of a graph is the maximum cardinality of its dissociation sets. In this paper, we study the connected graphs of order $n$ with a given dissociation number that attains the minimum spectral radius. We characterize these graphs when the dissociation number is in $\{n-1,~n-2,~\lceil2n/3\rceil,~\lfloor2n/3\rfloor,~2\}$. We also prove that these graphs are trees when the dissociation number is larger than $\lceil {2n}/{3}\rceil$.