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 Δ(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 Δ(G) if the product pq divides χ(1), for some χ∈Irr(G). Tong-Viet posed the conjecture that if Δ(G) is k-regular for some integer k⩾2, then Δ(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,∞).