On spectral extrema of graphs with given order and dissociation number

Abstract

Identifying the graph with maximum or minimum spectral radius among a given class of graphs is a central problem in extremal spectral graph theory, known as the Brualdi–Solheid problem. For a graph G=(VG,EG), a subset S⊆VG is called a maximum dissociation set if the induced subgraph G[S] does not contain P3 as its subgraph, and the subset has maximum cardinality. The dissociation number of G is the number of vertices in a maximum dissociation set of G. In this paper, we first characterize all the connected graphs (resp. bipartite graphs, trees) having maximum spectral radius among connected graphs (resp. bipartite graphs, trees) with given order and dissociation number. Secondly, we show that the connected n-vertex graph with dissociation number φ having minimum spectral radius is a tree, where φ≥23n. Finally, we determine the graphs having minimum spectral radius with fixed order n and dissociation number φ∈{2,⌈2n/3⌉,n−1,n−2}.