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Shrabani Das

Publications and source records attributed to Shrabani Das.

8 recordsLinked to original sources
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{\v{c}}evi{\'c} conjecture [Hansen, P. and Vuki{\v{c}}evi{\'c}, 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.

math.CO↗
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.

math.GR↗
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{\v{c}}evi{\'c} conjecture [Hansen, P. and Vuki{\v{c}}evi{\'c}, 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{\v{c}}evi{\'c} 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 $\deg_{\mathcal{B}(G)}(H)$ 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 $\mathcal{B}(G)$.

math.GR↗
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.

math.GR↗
arXiv2024

Commuting Conjugacy Class Graphs of Finite Groups and the Hansen-Vuki\v{c}evi\'c 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{\v{c}}evi{\'c} 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{\v{c}}evi{\'c} conjecture.

math.GR↗
arXiv2024

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{\v{c}}evi{\'c} conjecture.

math.GR↗
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 $\Gamma_c(G)$) 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.

math.GR↗