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Rajat Kanti Nath

Publications and source records attributed to Rajat Kanti Nath.

At least 19 recordsLinked to original sources
arXiv2026

Minimum second neighborhood degree energy of commuting graphs of finite rings

In this paper, we compute minimum second neighborhood degree spectrum and energy of commuting graphs of certain finite non-commutative rings. In particular, we consider non-commutative rings of order $p^2, p^3, p^4, p^5, p^2q$ and $p^3q$, where $p$ and $q$ are primes. We shall also show that the commuting graphs of these rings are MSN-integral but not MSN-hyperintegral. Finally, employing the techniques used in this paper, we prove Conjecture 3 of [Nath, R. K., Fasfous, W. N. T., Das, K. C. and Shang, Y. Common neighbourhood energy of commuting graphs of finite groups, {\em Symmetry} {\bf 13}(9), Article No. 1651, 2021.] and Conjecture 3.12 of [W. N. T. Fasfous and Nath, R. K. Common neighborhood spectrum and energy of commuting graphs of finite rings, \emph{ Palestine J. Math.} \textbf{13}(1), 66--76, 2024.]. We conclude this paper with two open problems.

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arXiv2026

Engel and co-Engel graphs of finite groups

Let $G$ be a group. Associate a directed graph $\vec{E}(G)$ (called the Engel digraph of $G$) with $G$ whose vertex set is $G$, with an arc $(x,y)$ if $[y, {}_k x]=1$ for some positive integer $k$, where $[y,{}_kx]$ is the iterated commutator $[y,x,x,\ldots,x]$, with $k$ terms $x$ in the expression. From this we define the Engel graph $E(G)$ by ignoring directions; the co-Engel graph $E_c(G)$ is its complement. The co-Engel graph, under the name ``Engel graph'', was introduced by Abdollahi. However, the name we use is more natural. We begin with some general results about the Engel digraph and graph, before turning our attention to the co-Engel graph. Among other things, we show that the undirected Engel graph does not determine the directed version up to isomorphism, though counterexamples seem to be fairly rare: there are just two orders less than $100$ for which this happens. We also prove a universality theorem: every finite digraph is an induced sub-digraph of the Engel digraph of a finite group. The isolated vertices of $E_c(G)$ form the Fitting subgroup $F(G)$ of $G$. In this paper, we realize the induced subgraph of co-Engel graphs of certain finite non-Engel groups $G$ induced by $G \setminus F(G)$. We write $E_c^-(G)$ to denote the subgraph of $E_c(G)$ induced by $G \setminus F(G)$. We also compute genus, various spectra, energies and Zagreb indices of $E_c^-(G)$ for those groups. As a consequence, we determine (up to isomorphism) all finite non-Engel group $G$ such that the clique number of $E_c^-(G)$ is at most $4$ and $E_c^-(G)$ is toroidal or projective. Further, we show that $E_c^-(G)$ is ALQ-integral and satisfies the E-LE conjecture and the Hansen-Vuki{č}evi{ć} conjecture for the groups considered in this paper.

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arXiv2026

Certain topological indices and spectral properties of SGB-graphs of finite cyclic groups

Let $L(G)$ be the set of all subgroups of a group $G$. The subgroup generating bipartite graph $\mathcal{B}(G)$ defined on $G$ is a bipartite graph whose vertex set is the union of two sets $G \times G$ and $L(G)$, and two vertices $(a, b) \in G \times G$ and $H \in L(G)$ are adjacent if $H$ is generated by $a$ and $b$. In this paper, we realize the structures of $\mathcal{B}(G)$ for cyclic groups of order $pq, p^2q$ and $p^2q^2$, where $p$ and $q$ are primes and $p \neq q$. We also deduce expressions for first and second Zagreb indices of these graphs and check the validity of Hansen-Vuki{č}evi{ć} conjecture [Hansen, P. and Vuki{č}evi{ć}, D. Comparing the Zagreb indices, {\em Croatica Chemica Acta}, \textbf{80}(2), 165-168, 2007]. Expressions of certain other degree-based topological indices of these graphs are also computed. We further compute various spectra and their corresponding energies of $\mathcal{B}(G)$ if $G$ is any cyclic group of order $p^n, pq, p^2q$ and $p^2q^2$, where $p$ and $q$ are two distinct primes and $n \geq 1$. We conclude the paper showing that $\mathcal{B}(G)$ satisfies E-LE conjecture [Gutman, I., Abreu, N. M. M., Vinagre, C. T. M., Bonifacioa, A. S. and Radenkovic, S. Relation between energy and Laplacian energy, {\em MATCH Communications in Mathematical and in Computer Chemistry}, \textbf{59}, 343--354, 2008] for these groups.

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arXiv2026

Various spectra and energies of subgroup generating bipartite graph

Let $L(G)$ be the set of all subgroups of a group $G$. The subgroup generating bipartite graph $\mathcal{B}(G)$ defined on $G$ is a bipartite graph whose vertex set is partitioned into two sets $G \times G$ and $L(G)$, and two vertices $(a, b) \in G \times G$ and $H \in L(G)$ are adjacent if $H$ is generated by $a$ and $b$. In this paper, we compute various spectra and energies of $\mathcal{B}(G)$ and determine whether $\mathcal{B}(G)$ is hypoenergetic, hyperenergetic, CN-hyperenergetic, L-hyperenergetic or Q-hyperenergetic if $G$ is a dihedral group of order $2p$ and $2p^2$ and dicyclic group of order $4p$ and $4p^2$, where $p$ is any prime. We also show that $\mathcal{B}(G)$ satisfies E-LE conjecture for these groups.

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arXiv2025

Super commuting graphs of finite groups and their Zagreb indices

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{č}evi{ć} conjecture.

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arXiv2025

Main functions and the spectrum of super graphs

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.

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arXiv2025

Various spectral aspects of NCCC-graphs of certain finite non-abelian groups

Let ${G}$ be a finite non-abelian group. The non-commuting conjugacy class graph (abbreviated as NCCC-graph) of $G$ is a simple undirected graph whose vertex set is the set of conjugacy classes of non-central elements of $G$ and two vertices $x^G$ and $y^G$ are adjacent to each other if $x'$ and $y'$ does not commute for all $x'\in x^G$ and $y'\in y^G$, where $x^G$ is the conjugacy class of $x \in G$. In this paper, we compute the spectrum, Laplacian spectrum, signless Laplacian spectrum and corresponding energies of NCCC-graphs of certain families of finite non-abelian groups. We determine whether these graphs are integral, L-integral and Q-integral. Further, we compare energy, Laplacian energy and signless Laplacian energy; and determine whether these graphs are borderenergetic, L-borderenergetic, Q-borderenergetic, hyperenergetic, L-hyperenergetic or Q-hyperenergetic.

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arXiv2025

Commuting Conjugacy Class Graphs of Finite Groups and the Hansen-Vukičević Conjecture

In this work, we compute the first and second Zagreb indices for the commuting conjugacy class graphs associated with finite groups. We identify multiple classes of finite groups whose commuting conjugacy class graphs are shown to satisfy the Hansen-Vuki{č}evi{ć} conjecture. Specifically, we prove that the conjecture holds for the commuting conjugacy class graphs of dihedral groups ($D_{2m}$), dicyclic groups, semidihedral groups, and various other two-generator groups. Moreover, we examine the case where the quotient $G/Z(G)$ is isomorphic to $D_{2m}$, $\mathbb{Z}_p \times \mathbb{Z}_p$, a Frobenius group of order $pq$ or $p^2q$, or any group of order $p^3$, for primes $p$ and $q$. In each of these cases, we demonstrate that the corresponding commuting conjugacy class graph satisfies the Hansen-Vuki{č}evi{ć} conjecture.

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arXiv2025

Zagreb indices of subgroup generating bipartite graph

Let $G$ be a group and $L(G)$ be the set of all subgroups of $G$. The subgroup generating bipartite graph $\mathcal{B}(G)$ defined on $G$ is a bipartite graph whose vertex set is partitioned into two sets $G \times G$ and $L(G)$, and two vertices $(a, b) \in G \times G$ and $H \in L(G)$ are adjacent if $H$ is generated by $a$ and $b$. In this paper, we deduce expressions for first and second Zagreb indices of $\mathcal{B}(G)$ and obtain a condition such that $\mathcal{B}(G)$ satisfy Hansen-Vuki{č}evi{ć} conjecture [Hansen, P. and Vuki{č}evi{ć}, D. Comparing the Zagreb indices, {\em Croatica Chemica Acta}, \textbf{80}(2), 165-168, 2007]. It is shown that $\mathcal{B}(G)$ satisfies Hansen-Vuki{č}evi{ć} conjecture if $G$ is a cyclic group of order $2p, 2p^2, 4p$, $4p^2$ and $p^n$; dihedral group of order $2p$ and $2p^2$; and dicyclic group of order $4p$ and $4p^2$ for any prime $p$. While computing Zagreb indices of $\mathcal{B}(G)$ we have computed $°_{\mathcal{B}(G)}(H)$ for all $H \in L(G)$ for the above mentioned groups. Using these information we also compute Randic Connectivity index, Atom-Bond Connectivity index, Geometric-Arithmetic index, Harmonic index and Sum-Connectivity index of $\mathcal{B}(G)$.

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arXiv2024

On a bipartite graph defined on groups

Let $G$ be a group and $L(G)$ be the set of all subgroups of $G$. We introduce a bipartite graph $\mathcal{B}(G)$ on $G$ whose vertex set is the union of two sets $G \times G$ and $L(G)$, and two vertices $(a, b) \in G \times G$ and $H \in L(G)$ are adjacent if $H$ is generated by $a$ and $b$. We establish connections between $\mathcal{B}(G)$ and the generating graph of $G$. We also discuss about various graph parameters such as independence number, domination number, girth, diameter, matching number, clique number, irredundance number, domatic number and minimum size of a vertex cover of $\mathcal{B}(G)$. We obtain relations between $\mathcal{B}(G)$ and certain probabilities associated to finite groups. We also obtain expressions for various topological indices of $\mathcal{B}(G)$. Finally, we realize the structures of $\mathcal{B}(G)$ for the dihedral groups of order $2p$ and $2p^2$ and dicyclic groups of order $4p$ and $4p^2$ (where $p$ is any prime) including certain other small order groups.

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arXiv2024

Common neighborhood (signless) Laplacian spectrum and energy of CCC-graph

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.

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arXiv2024

Common neighborhood Laplacian and signless Laplacian spectra and energies of commuting graphs

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.

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arXiv2024

Common neighborhood energies and their relations with Zagreb index

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.

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arXiv2024

Super graphs on groups, II

In an earlier paper, the authors considered three types of graphs, and three equivalence relations, defined on a group, viz.\ the power graph, enhanced power graph, and commuting graph, and the relations of equality, conjugacy, and same order; for each choice of a graph type A and an equivalence relation B, there is a graph, the \emph{B superA graph} defined on $G$. The resulting nine graphs (of which eight were shown to be in general distinct) form a two-dimensional hierarchy. In the present paper, we consider these graphs further. We prove universality properties for the conjugacy supergraphs of various types, adding the nilpotent, solvable and enhanced power graphs to the commuting graphs considered in the rest of the paper, and also examine their relation to the invariable generating graph of the group. We also show that supergraphs can be expressed as graph compositions, in the sense of Schwenk, and use this representation to calculate their Wiener index. We illustrate these by computing Wiener index of equality supercommuting and conjugacy supercommuting graphs for dihedral and quaternion groups.

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arXiv2024

Characterization of commuting graphs of finite groups having small genus

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 $Γ_c(G)$) is double-toroidal or triple-toroidal then $Γ_c(G)$ and its complement satisfy Hansen-Vuki{č}evi{ć} 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.

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arXiv2023

Signless Laplacian energies of non-commuting graphs of finite groups and related results

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.

math.GR↗
arXiv2022

Common neighbourhood spectrum and energy of commuting conjugacy class graph

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.

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