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Rajat Kanti Nath

Publications and source records attributed to Rajat Kanti Nath.

At least 37 records · Page 2Linked to original sources
arXiv2022

Solvable conjugacy class graph of groups

In this paper we introduce the graph $Γ_{sc}(G)$ 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 $d=2$.

math.CO↗
arXiv2021

Super graphs on groups, I

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.

math.CO↗
arXiv2021

On r-noncommuting graph of finite rings

Let $R$ be a finite ring and $r\in R$. The $r$-noncommuting graph of $R$, denoted by $Γ_R^r$, 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 $Γ_R^r$. We show that $Γ_R^r$ is not a regular graph, a lollipop graph and complete bipartite graph. Further, we consider an induced subgraph of $Γ_R^r$ (induced by the non-central elements of $R$) and obtained some characterizations of $R$.

math.RA↗
arXiv2021

Genus of commuting graphs of some classes of finite rings

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.

math.RA↗
arXiv2020

$g$-noncommuting graph of a finite group relative to its subgroups

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 $Δ_{H, G}^g$ 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 $Δ_{H, G}^g$ is a tree among other results. We also discuss about its diameter and connectivity with special attention to the dihedral groups.

math.GR↗
arXiv2020

Relative g-noncommuting graph of finite groups

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 $Γ_{H, G}^g$. In this paper, we obtain computing formulae for degree of any vertex in $Γ_{H, G}^g$ and characterize whether $Γ_{H, G}^g$ is a tree, star graph, lollipop or a complete graph together with some properties of $Γ_{H, G}^g$ involving isomorphism of graphs. We also present certain relations between the number of edges in $Γ_{H, G}^g$ and certain generalized commuting probabilities of $G$ which give some computing formulae for the number of edges in $Γ_{H, G}^g$. Finally, we conclude this paper by deriving some bounds for the number of edges in $Γ_{H, G}^g$.

math.GR↗
arXiv2020

Genus of commuting conjugacy class graph of groups

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.

math.GR↗
arXiv2020

Spectrum and energies of commuting conjugacy class graph of a finite group

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.

math.GR↗
arXiv2020

Spectral aspects of commuting conjugacy class graph of finite groups

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.

math.GR↗
arXiv2020

Spectrum and energy of non-commuting graphs of finite groups

Let $G$ be a finite non-abelian group and $Γ_{nc}(G)$ be its non-commuting graph. In this paper, we compute spectrum and energy of $Γ_{nc}(G)$ 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 $E(Γ_{nc}(G))$ and $LE(Γ_{nc}(G))$ respectively) of $Γ_{nc}(G)$ and conclude that $E(Γ_{nc}(G)) \leq LE(Γ_{nc}(G))$ 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.

math.GR↗
arXiv2019

Non-solvable graphs of groups

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.

math.GR↗
arXiv2019

A note on solvable graphs of finite groups

Let $G$ be a finite non-solvable group with solvable radical $Sol(G)$. The solvable graph $Γ_s(G)$ 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 $Γ_s (G)$ 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 $Γ_s (G)$ and derive a lower bound of the clique number of $Γ_s (G)$. 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 $Γ_s (G)$ and the solvability degree of $G$.

math.GR↗
arXiv2018

Commuting probabilities of $n$-centralizer finite rings

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.

math.RA↗
arXiv2017

On relative commuting probability of finite rings

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.

math.RA↗
arXiv2017

On relative autocommutativity degree of a subgroup of a finite group

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.

math.GR↗