Graphs for which the least eigenvalue is minimal, I

Abstract

Let G be a connected graph whose least eigenvalue λ(G) is minimal among the connected graphs of prescribed order and size. We show first that either G is complete or λ(G) is a simple eigenvalue. In the latter case, the sign pattern of a corresponding eigenvector determines a partition of the vertex set, and we study the structure of G in terms of this partition. We find that G is either bipartite or the join of two graphs of a simple form

On least eigenvalues of bicyclic graphs with fixed number of pendant vertices

“…. v r , r ≥ 2, in a graph G is called an internal path if these r vertices are distinct (except possibly v 1 …”

“…The least eigenvalue has received attention recently. Bell et al [1,2] established some properties of graphs for which the least eigenvalue of connected graphs of fixed order and size is minimal. Liu et al [12] determined the unique graph of unicyclic graph (connected graph with n vertices and n edges) of fixed order and number of pendant vertices for which the least eigenvalue is minimal.…”

Characterization of the Minimizing Graph of the Connected Graphs Whose Complements Are Bicyclic

“…For further study, we refer [12][13][14][15][16][17]. The rest of the paper is organized as follows: in Section 2, we present some basic definitions and terminologies that are frequently used in the main results and Section 3 includes the main results from the minimizing graph of the connected graphs whose complements are bicyclic.…”

A note on the least eigenvalue of a graph with given maximum degree

“…Then by the Rayleigh-Ritz Theorem, we have The research for the least eigenvalue of graphs in some class is well-studied and interesting. For example, Bell et al [1,2] studied the extremal graphs with n vertices…”