Abstract
Let G n , m represent the family of square power graphs of order n and size m , obtained from the family of graphs F n , k of order n and size k , with m ≥ k . In this paper, we discussed the least eigenvalue of graph G in the family G n , m c . All graphs considered here are undirected, simple, connected, and not a complete K n for positive integer n .
Highlights
We limit our discussion to only connected graphs because we know that the spectrum of a disconnected graph is the union of spectra of its connected components and so the least eigenvalue of a disconnected graph will be the minimum eigenvalue of its component among all the minimum eigenvalues of its components
If G is a bipartite graph of order n and size m, λmin(G) goes to its lower bound by increasing the size of G
Summary
We limit our discussion to only connected graphs because we know that the spectrum (the set of eigenvalues of a graph) of a disconnected graph is the union of spectra of its connected components and so the least eigenvalue of a disconnected graph will be the minimum eigenvalue of its component among all the minimum eigenvalues of its components.ere are many results in the literature regarding the least eigenvalue of a graph. We will denote the least eigenvalue of a graph G by λmin(G) and X for the corresponding least eigenvector. Hoffman [2] discussed the limiting point of the least eigenvalue of connected graphs.
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