Some new aspects of main eigenvalues of graphs

Abstract

An eigenvalue of the adjacency matrix of a graph is said to be main if the all-1 vector is non-orthogonal to the associated eigenspace. This paper explores some new aspects of the study of main eigenvalues of graphs, investigating specifically cones over strongly regular graphs and graphs for which the least eigenvalue is non-main. In this case, we characterize paths and trees with diameter-3 satisfying the property. We may note that the importance of least eigenvalues of graphs for the equilibria of social and economic networks was recently uncovered in literature.

Graphs with at most two nonzero distinct absolute eigenvalues

“…In this section we study connected graphs in G n that have exactly four distinct eigenvalues, i.e. with spectrum [λ] 1…”

“…where G is the complement of G. In addition, if G is connected and regular, then G ⊗ J m and G ⊛ J m are connected and regular. 1) . 1) .…”

“…Taking a graph G of order n in this family and an integer m ≥ 2, the graph G ⊗ J m is a irregular graph in G nm with four distinct eigenvalues such that 1) , −mq q 3 +q 2 +q .…”

“…1 ,[4] 7 , [0] 22 , [−4]14 . From each design graph in G n we can construct an infinite family of connected regular graphs with four distinct eigenvalues.…”