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Let $G$ be a group. Associate a graph $\mathcal{E}_G$ (called the co-Engel graph of $G$) with $G$ whose vertex set is $G$ and two distinct vertices $x$ and $y$ are adjacent if $[x, {}_k y] \neq 1$ and $[y, {}_k x] \neq 1$ for all positive integer $k$. This graph, under the name ``Engel graph'', was introduced by Abdollahi. Let $L(G)$ be the set of all left Engel elements of $G$. In this paper, we realize the induced subgraph of co-Engel graphs of certain finite non-Engel groups $G$ induced by $G \setminus L(G)$. We write $\mathcal{E}^-(G)$ to denote the subgraph of $\mathcal{E}_G$ induced by $G \setminus L(G)$. We also compute genus, various spectra, energies and Zagreb indices of $\mathcal{E}^-(G)$ for those groups. As a consequence, we determine (up to isomorphism) all finite non-Engel group $G$ such that the clique number is at most $4$ and $\mathcal{E}^-$ is toroidal or projective. Further, we show that $\coeng{G}$ is super integral and satisfies the E-LE conjecture and the Hansen--Vuki{\v{c}}evi{\'c} conjecture for the groups considered in this paper.

Let A be a graph type and B an equivalence relation on a group $G$. Let $[g]$ be the equivalence class of $g$ with respect to the equivalence relation B. The B superA graph of $G$ is an undirected graph whose vertex set is $G$ and two distinct vertices $g, h \in G$ are adjacent if $[g] = [h]$ or there exist $x \in [g]$ and $y \in [h]$ such that $x$ and $y$ are adjacent in the A graph of $G$. In this paper, we compute spectrum of equality/conjugacy supercommuting graphs of dihedral/dicyclic groups and show that these graphs are not integral.

Let $B$ be an equivalence relation defined on a finite group $G$. The $B$ super commuting graph on $G$ is a graph whose vertex set is $G$ and two distinct vertices $g$ and $h$ are adjacent if either $[g] = [h]$ or there exist $g' \in [g]$ and $h' \in [h]$ such that $g'$ commutes with $h'$, where $[g]$ is the $B$-equivalence class of $g \in G$. Considering $B$ as the equality, conjugacy and same order relations on $G$, in this article, we discuss the graph structures of equality/conjugacy/order super commuting graphs of certain well-known families of non-abelian groups viz. dihedral groups, dicyclic groups, semidihedral groups, quasidihedral groups, the groups $U_{6n}, V_{8n}, M_{2mn}$ etc. Further, we compute the Zagreb indices of these graphs and show that they satisfy Hansen-Vuki{\v{c}}evi{\'c} conjecture., Comment: 21 pages

In this paper, we consider commuting conjugacy class graph (abbreviated as CCC-graph) of a finite group $G$ which is a graph with vertex set $\{x^G : x \in G \setminus Z(G)\}$ (where $x^G$ denotes the conjugacy class containing $x$) and two distinct vertices $x^G$ and $y^G$ are joined by an edge if there exist some elements $x'\in x^G$ and $y'\in y^G$ such that they commute. We compute common neighborhood (signless) Laplacian spectrum and energy of CCC-graph of finite non-abelian groups whose central quotient is isomorphic to either $\mathbb{Z}_p \times \mathbb{Z}_p$ (where $p$ is any prime) or the dihedral group $D_{2n}$ ($n \geq 3$); and determine whether CCC-graphs of these groups are common neighborhood (signless) Laplacian hyperenergetic/borderenergetic. As a consequence, we characterize certain finite non-abelian groups viz. $D_{2n}$, $T_{4n}$, $U_{6n}$, $U_{(n, m)}$, $SD_{8n}$ and $V_{8n}$ such that their CCC-graphs are common neighborhood (signless) Laplacian hyperenergetic/borderenergetic., Comment: 31 pages

In this paper, we compute common neighbourhood Laplacian spectrum, common neighbourhood signless Laplacian spectrum and their respective energies of commuting graph of some finite non-abelian groups including some AC-groups, groups whose central quotients are isomorphic to $Sz(2)$, $\mathbb{Z}_p\times \mathbb{Z}_p$ or $D_{2m}$. Our findings lead us to conclude that these graphs are CNL (CNSL)-integral. Additionally, we characterize the aforementioned groups such that their commuting graphs are CNL (CNSL)-hyperenergetic., Comment: 36 pages

In this paper we establish connections between common neighborhood Laplacian and common neighborhood signless Laplacian energies and the first Zagreb index of a graph $\mathcal{G}$. We introduce the concepts of CNL-hyperenergetic and CNSL-hyperenergetic graphs and showed that $\mathcal{G}$ is neither CNL-hyperenergetic nor CNSL-hyperenergetic if $\mathcal{G}$ is a complete bipartite graph. We obtain certain relations between various energies of a graph. Finally, we conclude the paper with several bounds for common neighborhood Laplacian and signless Laplacian energies of a graph., Comment: 27 pages

In this paper we first show that among all double-toroidal and triple-toroidal finite graphs only $K_8 \sqcup 9K_1$, $K_8 \sqcup 5K_2$, $K_8 \sqcup 3K_4$, $K_8 \sqcup 9K_3$, $K_8\sqcup 9(K_1 \vee 3K_2)$, $3K_6$ and $3K_6 \sqcup 4K_4 \sqcup 6K_2$ can be realized as commuting graphs of finite groups. As consequences of our results we also show that for any finite non-abelian group $G$ if the commuting graph of $G$ (denoted by $\Gamma_c(G)$) is double-toroidal or triple-toroidal then $\Gamma_c(G)$ and its complement satisfy Hansen-Vuki{\v{c}}evi{\'c} Conjecture and E-LE conjecture. In the process we find a non-complete graph, namely the non-commuting graph of the group $(\mathbb{Z}_3 \times \mathbb{Z}_3) \rtimes Q_8$, that is hyperenergetic. This gives a new counter example to a conjecture of Gutman regarding hyperenergetic graphs., Comment: 16 pages

In this paper we compute first and second Zagreb indices of commuting and non-commuting graphs of finite groups and determine several classes of finite groups such that their commuting and non-commuting graphs satisfy Hansen-Vuki\v{c}evi\'c conjecture., Comment: 31 pages

The non-commuting graph of a non-abelian group $G$ with center $Z(G)$ is a simple undirected graph whose vertex set is $G\setminus Z(G)$ and two vertices $x, y$ are adjacent if $xy \ne yx$. In this study, we compute Signless Laplacian spectrum and Signless Laplacian energy of non-commuting graphs of finite groups. We obtain several conditions such that the non-commuting graph of $G$ is Q-integral and observe relations between energy, Signless Laplacian energy and Laplacian energy. In addition, we look into the energetic hyper- and hypo-properties of non-commuting graphs of finite groups. We also assess whether the same graphs are Q-hyperenergetic and L-hyperenergetic., Comment: 39 pages

In this paper we compute common neighbourhood (abbreviated as CN) spectrum and energy of commuting conjugacy class graph of several families of finite non-abelian groups. As a consequence of our results we show that the commuting conjugacy class graphs of the groups $D_{2n}$, $T_{4n}$, $SD_{8n}$, $U_{(n,m)}$, $U_{6n}$, $V_{8n}$, $G(p, m, n)$ and some families of groups whose central quotient is isomorphic to $D_{2n}$ or $Z_p \times Z_p$, for some prime $p$, are CN-integral but not CN-hyperenergetic., Comment: 20 pages

In this paper we introduce the graph $\Gamma_{sc}(G)$ associated with a group $G$, called the solvable conjugacy class graph (abbreviated as SCC-graph), whose vertices are the nontrivial conjugacy classes of $G$ and two distinct conjugacy classes $C, D$ are adjacent if there exist $x \in C$ and $y \in D$ such that $\langle x, y\rangle$ is solvable. We discuss the connectivity, girth, clique number, and several other properties of the SCC-graph. One of our results asserts that there are only finitely many finite groups whose SCC-graph has given clique number~$d$, and we find explicitly the list of such groups with $d=2$.

Let $G$ be a finite group. A number of graphs with the vertex set $G$ have been studied, including the power graph, enhanced power graph, and commuting graph. These graphs form a hierarchy under the inclusion of edge sets, and it is useful to study them together. In addition, several authors have considered modifying the definition of these graphs by choosing a natural equivalence relation on the group such as equality, conjugacy, or equal orders, and joining two elements if there are elements in their equivalence class that are adjacent in the original graph. In this way, we enlarge the hierarchy into a second dimension. Using the three graph types and three equivalence relations mentioned gives nine graphs, of which in general only two coincide; we find conditions on the group for some other pairs to be equal. These often define interesting classes of groups, such as EPPO groups, $2$-Engel groups, and Dedekind groups. We study some properties of graphs in this new hierarchy. In particular, we characterize the groups for which the graphs are complete, and in most cases, we characterize the dominant vertices (those joined to all others). Also, we give some results about universality, perfectness, and clique number. The paper ends with some open problems and suggestions for further work.

In this paper, we compute the genus of commuting graphs of non-commutative rings of order $p^4$, $p^5$, $p^2q$ and $p^3q$, where $p$ and $q$ are prime integers. We also characterize those finite rings such that their commuting graphs are planar or toroidal., Comment: 9 pages

Let $H$ be a subgroup of a finite non-abelian group $G$ and $g \in G$. Let $Z(H, G) = \{x \in H : xy = yx, \forall y \in G\}$. We introduce the graph $\Delta_{H, G}^g$ whose vertex set is $G \setminus Z(H, G)$ and two distinct vertices $x$ and $y$ are adjacent if $x \in H$ or $y \in H$ and $[x,y] \neq g, g^{-1}$, where $[x,y] = x^{-1}y^{-1}xy$. In this paper, we determine whether $\Delta_{H, G}^g$ is a tree among other results. We also discuss about its diameter and connectivity with special attention to the dihedral groups., Comment: 17 pages

Let $G$ be a finite group. For a fixed element $g$ in $G$ and a given subgroup $H$ of $G$, the relative $g$-noncommuting graph of $G$ is a simple undirected graph whose vertex set is $G$ and two vertices $x$ and $y$ are adjacent if $x \in H$ or $y \in H$ and $[x,y] \neq g, g^{-1}$. We denote this graph by $\Gamma_{H, G}^g$. In this paper, we obtain computing formulae for degree of any vertex in $\Gamma_{H, G}^g$ and characterize whether $\Gamma_{H, G}^g$ is a tree, star graph, lollipop or a complete graph together with some properties of $\Gamma_{H, G}^g$ involving isomorphism of graphs. We also present certain relations between the number of edges in $\Gamma_{H, G}^g$ and certain generalized commuting probabilities of $G$ which give some computing formulae for the number of edges in $\Gamma_{H, G}^g$. Finally, we conclude this paper by deriving some bounds for the number of edges in $\Gamma_{H, G}^g$., Comment: 23 pages

For a non-abelian group $G$, its commuting conjugacy class graph $\mathcal{CCC}(G)$ is a simple undirected graph whose vertex set is the set of conjugacy classes of the non-central elements of $G$ and two distinct vertices $x^G$ and $y^G$ are adjacent if there exists some elements $x' \in x^G$ and $y' \in y^G$ such that $x'y' = y'x'$. In this paper we compute the genus of $\mathcal{CCC}(G)$ for six well-known classes of non-abelian two-generated groups (viz. $D_{2n}, SD_{8n}, Q_{4m}, V_{8n}, U_{(n, m)}$ and $G(p, m, n)$) and determine whether $\mathcal{CCC}(G)$ for these groups are planar, toroidal, double-toroidal or triple-toroidal., Comment: 12 pages

In this paper we compute spectrum, Laplacian spectrum, signless Laplacian spectrum and their corresponding energies of commuting conjugacy class graph of the group $G(p, m, n) = \langle x, y : x^{p^m} = y^{p^n} = [x, y]^p = 1, [x, [x, y]] = [y, [x, y]] = 1\rangle$, where $p$ is any prime, $m \geq 1$ and $n \geq 1$. We derive some consequences along with the fact that commuting conjugacy class graph of $G(p, m, n)$ is super integral. We also compare various energies and determine whether commuting conjugacy class graph of $G(p, m, n)$ is hyperenergetic, L-hyperenergetic or Q-hyperenergetic., Comment: 13 pages

The commuting conjugacy class graph of a non-abelian group $G$, denoted by $\mathcal{CCC}(G)$, is a simple undirected graph whose vertex set is the set of conjugacy classes of the non-central elements of $G$ and two distinct vertices $x^G$ and $y^G$ are adjacent if there exists some elements $x' \in x^G$ and $y' \in y^G$ such that $x'y' = y'x'$. In this paper we compute various spectra and energies of commuting conjugacy class graph of the groups $D_{2n}, Q_{4m}, U_{(n, m)}, V_{8n}$ and $SD_{8n}$. Our computation shows that $\mathcal{CCC}(G)$ is super integral for these groups. We compare various energies and as a consequence it is observed that $\mathcal{CCC}(G)$ satisfy E-LE Conjecture of Gutman et al. We also provide negative answer to a question posed by Dutta et al. comparing Laplacian and Signless Laplacian energy. Finally, we conclude this paper by characterizing the above mentioned groups $G$ such that $\mathcal{CCC}(G)$ is hyperenergetic, L-hyperenergetic or Q-hyperenergetic., Comment: 36 pages

Let $G$ be a finite non-abelian group and ${\Gamma}_{nc}(G)$ be its non-commuting graph. In this paper, we compute spectrum and energy of ${\Gamma}_{nc}(G)$ for certain classes of finite groups. As a consequence of our results we construct infinite families of integral complete $r$-partite graphs. We compare energy and Laplacian energy (denoted by $E({\Gamma}_{nc}(G))$ and $LE({\Gamma}_{nc}(G))$ respectively) of ${\Gamma}_{nc}(G)$ and conclude that $E({\Gamma}_{nc}(G)) \leq LE({\Gamma}_{nc}(G))$ for those groups except for some non-abelian groups of order $pq$. This shows that the conjecture posed in [Gutman, I., Abreu, N. M. M., Vinagre, C. T.M., Bonifacioa, A. S and Radenkovic, S. Relation between energy and Laplacian energy, MATCH Commun. Math. Comput. Chem., 59: 343--354, (2008)] does not hold for non-commuting graphs of certain finite groups, which also produces new families of counter examples to the above mentioned conjecture., Comment: 22 pages

Let $G$ be a group and $Sol(G)=\{x \in G : \langle x,y \rangle \text{ is solvable for all } y \in G\}$. We associate a graph $\mathcal{NS}_G$ (called the non-solvable graph of $G$) with $G$ whose vertex set is $G \setminus Sol(G)$ and two distinct vertices are adjacent if they generate a non-solvable subgroup. In this paper we study many properties of $\mathcal{NS}_G$. In particular, we obtain results on vertex degree, cardinality of vertex degree set, graph realization, domination number, vertex connectivity, independence number and clique number of $\mathcal{NS}_G$. We also consider two groups $G$ and $H$ having isomorphic non-solvable graphs and derive some properties of $G$ and $H$. Finally, we conclude this paper by showing that $\mathcal{NS}_G$ is neither planar, toroidal, double-toroidal, triple-toroidal nor projective., Comment: 17 pages

Let $R$ be a finite ring and $r\in R$. The $r$-noncommuting graph of $R$, denoted by $\Gamma_R^r$, is a simple undirected graph whose vertex set is $R$ and two vertices $x$ and $y$ are adjacent if and only if $[x,y] \neq r$ and $-r$. In this paper, we study several properties of $\Gamma_R^r$. We show that $\Gamma_R^r$ is not a regular graph, a lollipop graph and complete bipartite graph. Further, we consider an induced subgraph of $\Gamma_R^r$ (induced by the non-central elements of $R$) and obtained some characterizations of $R$., Comment: 17 pages

Let $G$ be a finite non-solvable group with solvable radical $Sol(G)$. The solvable graph $\Gamma_s(G)$ of $G$ is a graph with vertex set $G\setminus Sol(G)$ and two distinct vertices $u$ and $v$ are adjacent if and only if $\langle u, v \rangle$ is solvable. We show that $\Gamma_s (G)$ is not a star graph, a tree, an $n$-partite graph for any positive integer $n \geq 2$ and not a regular graph for any non-solvable finite group $G$. We compute the girth of $\Gamma_s (G)$ and derive a lower bound of the clique number of $\Gamma_s (G)$. We prove the non-existence of finite non-solvable groups whose solvable graphs are planar, toroidal, double-toroidal, triple-toroidal or projective. We conclude the paper by obtaining a relation between $\Gamma_s (G)$ and the solvability degree of $G$., Comment: 10 pages

Let $R$ be a finite ring. The commuting probability of $R$, denoted by $\Pr(R)$, is the probability that any two randomly chosen elements of $R$ commute. $R$ is called an $n$-centralizer ring if it has $n$ distinct centralizers. In this paper, we compute $\Pr(R)$ for some $n$-centralizer finite rings., Comment: 9 pages

In this paper we study the probability that the commutator of a randomly chosen pair of elements, one from a subring of a finite ring and other from the ring itself equals to a given element of the ring.

Let $R$ be a finite ring and $r \in R$. The aim of this paper is to study the probability that the commutator of a randomly chosen pair of elements of $R$ equals $r$.

In this paper, we consider the probability that a randomly chosen automorphism of a finite group fixes a randomly chosen element of a subgroup of that group. We obtain several new results as well as generalizations and improvements of some existing results on this probability.

In this paper, we compute Laplacian energy of the non-commuting graphs of some classes of finite non-abelian groups., Comment: arXiv admin note: substantial text overlap with arXiv:1705.01275; text overlap with arXiv:1705.00137