The Least Distance Eigenvalue of Complement Graphs with Two Pendant Vertices

Abstract

Let G be a connected simple graph with the vertex set V(G)={v1,v2,…,vn}, where the distance dG(vi,vj) is the length of a shortest path between vi and vj, and the distance matrix D(G)=(dij)n×n is defined by dij=dG(vi,vj). Being real symmetric and non-negative, D(G) has real eigenvalues λ1(G)≥λ2(G)≥…≥λn(G), with λn(G) referred to as the least distance eigenvalue of G. In this paper, we give the maximum value of the least distance eigenvalue, and determine the unique extremal graph with two pendant vertices whose complements attain the value.