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Let $G$ be a finite group, and let $\mathrm{Irr}(G)$ denote the set of irreducible complex characters of $G$. An element $x$ of $G$ is said to be vanishing, if for some $\chi$ in $\mathrm{Irr}(G)$, we have $\chi(x)=0$. Also the element $x$ is called rational if $x$ is conjugate to $x^i$ for every integer $i$ co-prime to the order of $x$. We define the weight of $G$ as $\omega(G):=(\sum_{\chi\in \mathrm{Irr}(G)}\chi(1))^2/|G|$. In this paper, we show that for every rational non-vanishing element $x\in G$, the order of $C_G(x)$ is at least $\omega(G)$.

Let $\Gamma$ be a simple graph with $n$ vertices. The energy of $\Gamma$, denoted by $\mathcal{E}(\Gamma)$, is defined as the sum of the absolute values of the eigenvalues of $\Gamma$. The graph $\Gamma$ is said to be hyperenergetic if $\mathcal{E}(\Gamma)>2n-2$. For the graph $\Gamma$, the multiplicity of the eigenvalue $0$, denoted by $\eta(\Gamma)$, is called the nullity of $\Gamma$. In this paper, we show that for every positive integer $n\geq 4$, the Cayley graph $\Gamma_n$ on the symmetric group $\mathrm{Sym}(n)$ generated by $n$-cycles is an integral hyperenergetic graph with $\mathcal{E}(\Gamma_n)=2^{n-1}(n-1)!$ and $\eta(\Gamma_n)=n!-\binom{2n-2}{n-1}$., Comment: arXiv admin note: text overlap with arXiv:1808.06097 by other authors

For a simple graph $G$, the complement and the line graph of $G$ are denoted by $G^c$ and $L(G)$, respectively. In this paper, we show that for every simple connected regular graph $G$ with at least $5$ vertices, the graph $\mathcal{R}(G):=L(L(G)^c)^c$ is a Ramanujan graph with diameter at most three.

For a graph $\Gamma$, the multiplicity of the eigenvalue $0$, denoted by $\eta(\Gamma)$, is called the nullity of $\Gamma$. Also the energy of $\Gamma$, denoted by $\mathcal{E}(\Gamma)$, is defined as the sum of the absolute values of the eigenvalues of $\Gamma$. The index of a subgroup $H$ in a group $G$ is denoted by $[G:H]$. For a prime $p$, let $G$ be a finite $p$-solvable group whose order is divisible by $p$. Also let $\Omega_p(G)$ be the set of all $p$-singular elements of $G$. In this paper, we apply block theory of finite groups to show that the Cayley graph $\Gamma_p(G):=\mathrm{Cay}(G,\Omega_p(G))$ is an integral graph with $\eta(\Gamma_p(G))=|G|-[G:O_{p^\prime}(G)]$, where $O_{p^\prime}(G)$ is the largest normal subgroup of $G$ whose order is co-prime to $p$. We also find a lower bound for $\mathcal{E}(\Gamma_p(G))$. Finally, we prove that the diameter of $\Gamma_p(G)$ is at most $ |G|_p$.

A transversal matroid whose dual is also transversal is called bi-transversal. Let $G$ be an undirected graph with vertex set $V$. In this paper, for every subset $W$ of $V$, we associate a bi-transversal matroid to the pair $(G,W)$. We also derive an explicit formula for counting bases of this matroid.

We study subsets $T$ consisting of some transpositions $(i,j)$ of the symmetric group $S_n$ on $\{1,\dots,n\}$ such that the Cayley graph $\Gamma_T:=Cay(S_n,T)$ is an integral graph, i.e., all eigenvalues of an adjacency matrix of $\Gamma_T$ are integers. Graph properties of $\Gamma_T$ are determined in terms of ones of the graph $G_T$ whose vertex set is $\{1,\dots,n\}$ and $\{i,j\}$ is an edge if and only if $(i,j)\in T$. Here we prove that if $G_T$ is a tree then $\Gamma_T$ is integral if and only if $T$ is isomorphic to the star graph $K_{1,n-1}$, answering Problem 5 of [Electron. J. Comnin., 29(2) (2022) \# P2.9]. Problem 6 of the latter article asks to find necessary and sufficient conditions on $T$ for integralness of $Cay(S_n,T)$ without any further assumption on $T$. We show that if $G_T$ is a graph which we call it a ``generalized complete multipartite graph" then $Cay(S_n,T)$ is integral. We conjecture that $Cay(S_n,T)$ is integral only if $G_T$ is a generalized complete multipartitie graph. To support the latter conjecture we show its validity whenever $G_T$ is some classes of graphs including cycles and cubic graphs., Comment: 8 pages

We first describe a system of inequalities (Horn's inequalities) that characterize eigenvalues of sums of Hermitian matrices. When we apply this system for integral Hermitian matrices, one can directly test it by using Littlewood-Richardson coefficients. In this paper, we apply Horn's inequalities to analysis the eigenvalues of an integral line graph $G$ of a connected bipartite graph. Then we show that the diameter of $G$ is at most $2\omega(G)$, where $\omega(G)$ is the clique number of $G$. Also using Horn's inequalities, we show that for every odd integer $k\geq 19$, a non-complete $k$-regular Ramanujan graph has an eigenvalue less than $-2$., Comment: arXiv admin note: text overlap with arXiv:math/9908012 by other authors

Let $S_n$ denote the symmetric group on $n$ letters. The $k$-point fixing graph $\mathcal{F}(n,k)$ is defined to be the graph with vertex set $S_n$ and two vertices $g,h$ of $\mathcal{F}(n,k)$ are joined by an edge, if and only if $gh^{-1}$ fixes exactly $k$ points. Ku, Lau and Wong [Cayley graph on symmetric group generated by elements fixing $k$ points, Linear Algebra Appl. 471 (2015) 405-426] obtained a recursive formula for the eigenvalues of $\mathcal{F}(n,k)$. In this paper, we use objects called excited diagrams defined as certain generalizations of skew shapes and derive an explicit formula for the eigenvalues of Cayley graph $\mathcal{F}(n,k)$. Then we apply this formula and show that the eigenvalues of $\mathcal{F}(n,k)$ are in the interval $[\frac{-|S(n,k)|}{n-k-1}, |S(n,k)|]$, where $S(n,k)$ is the set of elements $\sigma$ of $S_n$ such that $\sigma$ fixes exactly $k$ points.

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 $[-2, \infty )$.

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 $S_3$.

Let $G$ be a finite group and let $\rm{Irr}(G)$ be the set of all irreducible complex characters of $G$. Let $\rm{cd}(G)$ be the set of all character degrees of $G$ and denote by $\rho(G)$ the set of primes which divide some character degrees in $\rm{cd}(G)$. The character 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 $G$. Suppose the character graph $\Delta(G)$ is $K_4$-free with diameter $3$. In this paper, we show that $|\rho(G)|\neq 5$, if and only if $G\cong J_1 \times A$, where $J_1$ is the first Janko's sporadic simple group and $A$ is abelian.

Suppose $\Gamma$ is a finite simple graph. If $D$ is a dominating set of $\Gamma$ such that each $x\in D$ is contained in the set of vertices of an odd cycle of $\Gamma$, then we say that $D$ is an odd dominating set for $\Gamma$. 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 show that the complement of $\Delta(G)$ contains an odd dominating set, if and only if $\Delta(G)$ is a disconnected graph with non-bipartite complement., Comment: arXiv admin note: text overlap with arXiv:2002.01353, arXiv:1909.01180, arXiv:1909.03062

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

Let $\Gamma$ be a finite simple graph. If for some integer $n\geqslant 4$, $\Gamma$ is a $K_n$-free graph whose complement has an odd cycle of length at least $2n-5$, then we say that $\Gamma$ is an $n$-exact graph. 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 prove that the order of an $n$-exact character graph is at most $2n-1$. Also we determine the structure of all finite groups $G$ with extremal $n$-exact character graph $\Delta(G)$., Comment: arXiv admin note: substantial text overlap with arXiv:1909.01180, arXiv:1909.03062, arXiv:1907.13292

L\'evai and Pyber proposed the following as a conjecture: Let $G$ be a profinite group such that the set of solutions of the equation $x^n=1$ has positive Haar measure. Then $G$ has an open subgroup $H$ and an element $t$ such that all elements of the coset $tH$ have order dividing $n$ (see Problem 14.53 of [The Kourovka Notebook, No. 19, 2019]). The validity of the conjecture has been proved in [Arch. Math. (Basel) 75 (2000) 1-7] for $n=2$. Here we study the conjecture for compact groups $G$ which are not necessarily profinite and $n=3$; we show that in the latter case the group $G$ contains an open normal $2$-Engel subgroup.

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 show that if the diameter of $\Delta(G)$ is equal to three, then the complement of $\Delta(G)$ is bipartite. Also in this case, we determine the structure of the character graph $\Delta(G)$., Comment: arXiv admin note: substantial text overlap with arXiv:1909.01180; text overlap with arXiv:1907.13292

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.

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 graph theory, a perfect graph is a graph $\Gamma$ in which the chromatic number of every induced subgraph $\Delta$ of $\Gamma$ equals the clique number of $\Delta$. In this paper, we show that the character graph $\Delta(G)$ of a finite group $G$ is always a perfect graph. We also prove that the chromatic number of the complement of $\Delta(G)$ is at most three.