Regular character-graphs whose eigenvalues are greater than or equal to -2

Abstract

Let G be a finite group and Irr(G) be the set of all complex irreducible characters of G. The character-graph ∆(G) associated to G, is a graph whose vertex set is the set of primes which divide the degrees of some characters in Irr(G) and two distinct primes p and q are adjacent in ∆(G) if the product pq divides χ(1), for some χ ∈ Irr(G). Tong-viet posed the conjecture that if ∆(G) is k-regular for some integer k 2, then ∆(G) is either a complete graph or a cocktail party graph. In this paper, we show that his conjecture is true for all regular character-graphs whose eigenvalues are in the interval [−2, ∞).