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For a digraph $\Gamma$, a subset $C$ of $V(\Gamma)$ is a perfect code if $C$ is a dominating set such that every vertex of $\Gamma$ is dominated by exactly one vertex in $C$. In this paper, we classify strongly connected 2-valent Cayley digraphs on abelian groups admitting a perfect code, and determine completely all perfect codes of such digraphs., Comment: arXiv admin note: text overlap with arXiv:2207.06743

Given a finite group $G$, the commuting graph of $G$ is the simple graph whose vertex set is $G$, and two distinct vertices are adjacent if they commute. In this paper, we classify all finite groups whose commuting graph is split and threshold. We also investigate the structure of a group whose commuting graph is either a cograph or a chordal graph, and determine all non-abelian finite simple groups whose commuting graph is a cograph. The results partly answer a question by Peter J. Cameron established in 2022 in his survey paper on graphs defined on groups., Comment: 26 pages

A subset $C$ of the vertex set of a graph $\Gamma$ is called a perfect code of $\Gamma$ if every vertex of $\Gamma$ is at distance no more than one to exactly one vertex in $C$. In this paper, we classify all connected quintic Cayley graphs on abelian groups that admit a perfect code, and determine completely all perfect codes of such graphs.

The enhanced power graph of a finite group $G$, denoted by $\mathcal{P}_E(G)$, is the simple undirected graph whose vertex set is $G$ and two distinct vertices $x, y$ are adjacent if $x, y \in \langle z \rangle$ for some $z \in G$. In this article, we determine all finite groups such that the minimum degree and the vertex connectivity of $\mathcal{P}_E(G)$ are equal. Also, we classify all groups whose (proper) enhanced power graphs are strongly regular. Further, the vertex connectivity of the enhanced power graphs associated to some nilpotent groups is obtained. Finally, we obtain a lower bound and an upper bound for the Wiener index of $\mathcal{P}_E(G)$, where $G$ is a nilpotent group. The finite nilpotent groups attaining these bounds are also characterized., Comment: arXiv admin note: text overlap with arXiv:2207.04641

We determine the diameter of every connected component of the complement of the power graph and the enhanced power graph of a finite group, which completely answers two questions by Peter J. Cameron., Comment: This paper has been accepted by the journal Ars Mathematica Contemporanea

The enhanced power graph of a group is the simple graph whose vertex set is consisted of all elements of the group, and whose any pair of vertices are adjacent if they generate a cyclic subgroup. In this paper, we classify all finite groups whose enhanced power graphs are split and threshold. We also classify all finite nilpotent groups whose enhanced power graphs are chordal graphs and cographs. Finally, we give some families of non-nilpotent groups whose enhanced power graphs are chordal graphs and cographs. These results partly answer a question posed by Peter J. Cameron., Comment: This paper has been accepted for publication in Revista de la Real Academia de Ciencias Exactas, F\'isicas y Naturales. Serie A. Matem\'aticas. Note that this is not a finally version. Its finally version will soon appear in the journal's homepage

Given a connected graph $G=(V(G), E(G))$, the length of a shortest path from a vertex $u$ to a vertex $v$ is denoted by $d(u,v)$. For a proper subset $W$ of $V(G)$, let $m(W)$ be the maximum value of $d(u,v)$ as $u$ ranging over $W$ and $v$ ranging over $V(G)\setminus W$. The proper subset $W=\{w_1,\ldots,w_{|W|}\}$ is a {\em completeness-resolving set} of $G$ if $$ \Psi_W: V(G)\setminus W \longrightarrow [m(W)]^{|W|},\qquad u\longmapsto (d(w_1,u),\ldots,d(w_{|W|},u)) $$ is a bijection, where $$ [m(W)]^{|W|}=\{(a_{(1)},\ldots,a_{(|W|)})\mid 1\leq a_{(i)}\leq m(W)\text{ for each }i=1,\ldots,|W|\}. $$ A graph is {\em completeness-resolvable} if it admits a completeness-resolving set. In this paper, we first construct the set of all completeness-resolvable graphs by using the edge coverings of some vertices in given bipartite graphs, and then establish posets on some subsets of this set by the spanning subgraph relationship. Based on each poset, we find the maximum graph and give the lower and upper bounds for the number of edges in a minimal graph. Furthermore, minimal graphs satisfying the lower or upper bound are characterized., Comment: 20 pages

Let $G$ be a finite group. The co-prime order graph of $G$ is the graph whose vertex set is $G$, and two distinct vertices $x,y$ are adjacent if gcd$(o(x),o(y))$ is either $1$ or a prime, where $o(x)$ and $o(y)$ are the orders of $x$ and $y$, respectively. In this paper, we characterize all finite groups whose co-prime order graphs are complete and classify all finite groups whose co-prime order graphs are planar. Also, we compute the vertex-connectivity of the co-prime order graph of a cyclic group, a dihedral group and a generalized quaternion group, which answers a question by Banerjee (2019). Finally, we prove that, for a fixed positive integer $k$, there are finitely many finite groups whose co-prime order graphs have (non)orientable genus $k$. As applications, we classify all finite groups whose co-prime order graphs have (non)orientable genus one and two., Comment: There is an error in Sect. 4. We can not obtain the first main result of this section

A subset $C$ of the vertex set of a graph $\Gamma$ is called a perfect code of $\Gamma$ if every vertex of $\Gamma$ is at distance no more than one to exactly one vertex in $C$. Let $A$ be a finite abelian group and $T$ a square-free subset of $A$. The Cayley sum graph of $A$ with respect to the connection set $T$ is a simple graph with $A$ as its vertex set, and two vertices $x$ and $y$ are adjacent whenever $x+y\in T$. A subgroup of $A$ is said to be a subgroup perfect code of $A$ if the subgroup is a perfect code of some Cayley sum graph of $A$. In this paper, we give some necessary and sufficient conditions for a subset of $A$ to be a perfect code of a given Cayley sum graph of $A$. We also characterize all subgroup perfect codes of $A$., Comment: 13 pp

Let $G$ be a finite group. The order supergraph of $G$ is the graph with vertex set $G$, and two distinct vertices $x,y$ are adjacent if $o(x)\mid o(y)$ or $o(y)\mid o(x)$. The enhanced power graph of $G$ is the graph whose vertex set is $G$, and two distinct vertices are adjacent if they generate a cyclic subgroup. The reduced power graph of $G$ is the graph with vertex set $G$, and two distinct vertices $x,y$ are adjacent if $\langle x\rangle \subset \langle y\rangle$ or $\langle y\rangle \subset \langle x\rangle$. In this paper, we characterize the strong metric dimension of the order supergraph, the enhanced power graph and the reduced power graph of a finite group., Comment: This is the final version to be published in Communications in Algebra, 16 pages

Let $\Gamma$ be a graph with vertex set $V(\Gamma)$. A subset $C$ of $V(\Gamma)$ is called a perfect code in $\Gamma$ if $C$ is an independent set of $\Gamma$ and every vertex in $V(\Gamma)\setminus C$ is adjacent to exactly one vertex in $C$. A subset $C$ of a group $G$ is called a perfect code of $G$ if there exists a Cayley graph of $G$ which admits $C$ as a perfect code. A group $G$ is said to be code-perfect if every proper subgroup of $G$ is a perfect code of $G$. In this paper we prove that a group is code-perfect if and only if it has no elements of order $4$. We also prove that a proper subgroup $H$ of an abelian group $G$ is a perfect code of $G$ if and only if the Sylow $2$-subgroup of $H$ is a perfect code of the Sylow $2$-subgroup of $G$. This reduces the problem of determining when a given subgroup of an abelian group is a perfect code to the case of abelian $2$-groups. Finally, we determine all subgroup perfect codes in any generalized quaternion group., Comment: This is the final version to be published in SIAM Journal on Discrete Mathematics, 18 pp

The power graph $\Gamma_G$ of a finite group $G$ is the graph with the vertex set $G$, where two distinct elements are adjacent if one is a power of the other. An $L(2, 1)$-labeling of a graph $\Gamma$ is an assignment of labels from nonnegative integers to all vertices of $\Gamma$ such that vertices at distance two get different labels and adjacent vertices get labels that are at least $2$ apart. The lambda number of $\Gamma$, denoted by $\lambda(\Gamma)$, is the minimum span over all $L(2, 1)$-labelings of $\Gamma$. In this paper, we obtain bounds for $\lambda(\Gamma_G)$, and give necessary and sufficient conditions when the bounds are attained. As applications, we compute the exact value of $\lambda(\Gamma_G)$ if $G$ is a dihedral group, a generalized quaternion group, a $\mathcal{P}$-group or a cyclic group of order $pq^n$, where $p$ and $q$ are distinct primes and $n$ is a positive integer., Comment: 13 pages, 1 figure

The {\em power index} $\Theta(\Gamma)$ of a graph $\Gamma$ is the least order of a group $G$ such that $\Gamma$ can embed into the power graph of $G$. Furthermore, this group $G$ is {\em $\Gamma$-optimal} if $G$ has order $\Theta(\Gamma)$. We say that $\Gamma$ is {\em power-critical} if its order equals to $\Theta(\Gamma)$. This paper focuses on the power indices of complete graphs, complete bipartite graphs and $1$-factors. We classify all power-critical graphs $\Gamma'$ in these three families, and give a necessary and sufficient condition for $\Gamma'$-optimal groups., Comment: 12 pages, 1 figure

We characterize the strong metric dimension of the power graph of a finite group. As applications, we compute the strong metric dimension of the power graph of a cyclic group, an abelian group, a dihedral group or a generalized quaternion group., Comment: 11 pages

For a finite noncyclic group $G$, let $\Cyc(G)$ be a set of elements $a$ of $G$ such that $\langle a,b\rangle$ is cyclic for each $b$ of $G$. The noncyclic graph of $G$ is a graph with the vertex set $G\setminus \Cyc(G)$, having an edge between two distinct vertices $x$ and $y$ if $\langle x, y\rangle$ is not cyclic. In this paper, we classify all finite noncyclic groups whose noncyclic graphs are $K_{1,n}$-free, where $3\leq n\leq 6$., Comment: 10 pages

Let $G$ be a finite non-cyclic group. The non-cyclic graph $\Gamma_G$ of $G$ is the graph whose vertex set is $G\setminus Cyc(G)$, two distinct vertices being adjacent if they do not generate a cyclic subgroup, where $Cyc(G)=\{a\in G: \langle a,b\rangle\ \text{is cyclic for each } b\in G\}$. In this paper, we classify all finite non-cyclic groups $G$ such that $\Gamma_G$ has (non)orientable genus one., Comment: 9 pages, 3 figures

The power graph $\Gamma_G$ of a finite group $G$ is the graph whose vertex set is the group, two distinct elements being adjacent if one is a power of the other. In this paper, we classify the finite groups whose power graphs have (non)orientable genus two., Comment: 17 pages, 7 figures

We give the characteristic polynomial of the distance or adjacency matrix of the strong power graph of a finite group, and compute its distance and adjacency spectrum., Comment: 6 pages

Let $G$ be a finite non-abelian group. The non-commuting graph $\Gamma_G$ of $G$ has the vertex set $G\setminus Z(G)$ and two distinct vertices $x$ and $y$ are adjacent if $xy\ne yx$, where $Z(G)$ is the center of $G$. We prove that the rainbow $2$-connectivity of $\Gamma_G$ is $2$. In particular, the rainbow connection number of $\Gamma_G$ is $2$. Moreover, for any positive integer $k$, we prove that there exist infinitely many non-abelian groups $G$ such that the rainbow $k$-connectivity of $\Gamma_G$ is $2$., Comment: 8 pages

This paper studies the rainbow connection number of the power graph $\Gamma_G$ of a finite group $G$. We determine the rainbow connection number of $\Gamma_G$ if $G$ has maximal involutions or is nilpotent, and show that the rainbow connection number of $\Gamma_G$ is at most three if $G$ has no maximal involutions. The rainbow connection numbers of power graphs of some nonnilpotent groups are also given., Comment: 10 pages, 5 figures, accepted by Graphs and Combinatorics on July 2015