Graphs having two main eigenvalues and arbitrarily many distinct vertex degrees

Abstract

Arif, Hayat and Khan [J Appl Math Comput 69 (2023) 2549–2571] recently proposed the problem of finding explicit construction for (an infinite family of) graphs having at least three distinct vertex degrees and two main eigenvalues. After computationally identifying small examples of such graphs, we fully solve this problem by showing that the edge-disjoint union of an almost semiregular graph G and a regular graph H defined on the constant part of G yields a new harmonic graph under mild conditions. As a special case, this result provides for every integer $b\ge 2$ an explicit construction of a graph with two main eigenvalues and $2b-1$ distinct vertex degrees. This construction also provides partial answers to questions posed by Hayat et al. in [Linear Algebra Appl 511 (2016) 318–327].