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Let $G$ be a finite group and let $\rm{Irr}(G)$ be the set of all irreducible complex characters of $G$. For a character $\chi \in \rm{Irr}(G)$, the number $\rm{cod}(\chi):=|G:\rm{ker}\chi|/\chi(1)$ is called the co-degree of $\chi$. The set of co-degrees of all irreducible characters of $G$ is denoted by $\rm{cod}(G)$. In this paper, we show that for a non-trivial finite group $G$, $|\rm{Irr}(G)|=|\rm{cod}(G)|$ if and only if $G$ is isomorphic to the cyclic group $\mathbb{Z}_2$ or the symmetric group $S_3$.

For a finite group $G$, let $\Delta(G)$ denote the character graph built on the set of degrees of the irreducible complex characters of $G$. Akhlaghi and Tong-Viet in \cite{[AT]} conjectured that if for some positive integer $n$, $\Delta(G)$ is $K_n$-free, then $\Delta(G)$ has at most $2n-1$ vertices. In this paper, we present an example to show that this conjecture is not necessarily true for all non-solvable groups whose character graphs are $K_n$-free., Comment: arXiv admin note: text overlap with arXiv:2002.01353. The counterexample given in this manuscripts has been presented in [Bounding the number of vertices in the degree graph of a finite group, Z. Akhlaghi, S. Dolfi, E. Pacifici, L. Sanus, Journal of Pure and Applied Algebra, 224(2020) 725-731

For a finite group $G$, let $\Delta(G)$ denote the character graph built on the set of degrees of the irreducible complex characters of $G$. In this paper, we determine the structure of all finite groups $G$ with $K_4$-free character graph $\Delta(G)$ having seven vertices. We also obtain a classification of all $K_4$-free graphs with seven vertices which can occur as character graphs of some finite groups.

Let G be a finite group and let Irr(G) be the set of all irreducible complex characters of G. Let cd(G) be the set of all character degrees of G and denote by \rho(G) the set of primes which divide some character degrees of G. The prime graph \Delta(G) associated to G is a graph whose vertex set is \rho(G) and there is an edge between two distinct primes p and q if and only if the product pq divides some character degree of G. In this paper, we show that the prime graph \Delta(G) of a finite group G is 3-regular if and only if it is a complete graph with four vertices., Comment: 18 pages

Let G be a finite group and let cd(G) be the set of all complex irreducible character degrees of G Let \rho(G) be the set of all primes which divide some character degree of G. The prime graph \Delta(G) attached to G is a graph whose vertex set is \rho(G) and there is an edge between two distinct primes u and v if and only if the product uv divides some character degree of G. In this paper, we show that if G is a finite group whose prime graph \Delta(G) has no triangles, then \Delta(G) has at most 5 vertices. We also obtain a classification of all finite graphs with 5 vertices and having no triangles which can occur as prime graphs of some finite groups. Finally, we show that the prime graph of a finite group can never be a cycle nor a tree with at least 5 vertices., Comment: 13 pages