The Optimal Graph Whose Least Eigenvalue is Minimal among All Graphs via 1-2 Adjacency Matrix

Abstract

All graphs under consideration are finite, simple, connected, and undirected. Adjacency matrix of a graph G is 0,1 matrix A=aij=0, if vi=vj or dvi,vj≥21, if dvi,vj=1.. Here in this paper, we discussed new type of adjacency matrix known by 1-2 adjacency matrix defined as A1,2G=aij=0, if vi=vj or dvi,vj≥31, if dvi,vj=2, from eigenvalues of the graph, we mean eigenvalues of the 1-2 adjacency matrix. Let Tnc be the set of the complement of trees of order n. In this paper, we characterized a unique graph whose least eigenvalue is minimal among all the graphs in Tnc.