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

TL;DR

This paper proves Tong-viet's conjecture for regular character-graphs with eigenvalues greater than or equal to -2, showing they are either complete graphs or cocktail party graphs.

Contribution

It extends the proof of Tong-viet's conjecture to a class of character-graphs with eigenvalues in [-2, ∞), confirming their structural classification.

Findings

Regular character-graphs with eigenvalues ≥ -2 are either complete or cocktail party graphs.

The conjecture holds for all such graphs with eigenvalues in the specified interval.

Eigenvalue conditions are crucial for classifying the structure of these graphs.

Abstract

Let $G$ be a finite group and $\mathrm{Irr}(G)$ be the set of all complex irreducible characters of $G$. The character-graph $\Delta(G)$ associated to $G$, is a graph whose vertex set is the set of primes which divide the degrees of some characters in $\mathrm{Irr}(G)$ and two distinct primes $p$ and $q$ are adjacent in $\Delta(G)$ if the product $pq$ divides $\chi(1)$, for some $\chi\in\mathrm{Irr}(G)$. Tong-viet posed the conjecture that if $\Delta(G)$ is $k$-regular for some integer $k\geqslant 2$, then $\Delta(G)$ 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 $[-2, \infty )$.