arXiv subjects

R. Rajkumar

Publications and source records attributed to R. Rajkumar.

At least 19 recordsLinked to original sources

arXiv2026

Adjacency-diametrical matrix of a graph

The adjacency-diametrical matrix (AD matrix) of a connected graph $G$ with diameter $d$, denoted by $AD(G)$, is the matrix indexed by the vertices of $G$ in which the $(i,j)$-entry of $AD(G)$ is $1$ if $d_G(v_i,v_j)=1$, is $d$ if $d_G(v_i,v_j)=d$, and $0$ otherwise, where $d_G(v_i,v_j)$ denotes the distance between the vertices $v_i$ and $v_j$ in $G$. We determine the spectrum of the AD matrix for paths, cycles, and double star graphs and obtain its determinant for a connected graph. We characterize a class of bipartite graphs using the coefficients of the characteristic polynomial and the eigenvalues of the AD matrix. We establish bounds relating the eigenvalues of the AD matrix to various graph invariants, and we determine the spectrum of the AD matrix for graphs formed by the join, lexicographic product, and Cartesian product operations under certain conditions on the constituent graphs.

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arXiv2025

New matrices for the spectral theory of mixed graphs, Part II

The concept of the integrated adjacency matrix for mixed graphs was first introduced in [9], where its spectral properties were analyzed in relation to the structural characteristics of the mixed graph. Building upon this foundation, this paper introduces the integrated Laplacian matrix, the integrated signless Laplacian matrix, and the normalized integrated Laplacian matrix for mixed graphs. We further explore how the spectra of these matrices relate to the structural properties of the mixed graph.

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arXiv2024

New matrices for the spectral theory of mixed graphs, part I

In this paper, we introduce a matrix for a mixed graph, called the integrated adjacency matrix. This matrix uniquely determines a mixed graph, as long as the indices of the matrix are specified. Additionally, we associate an (undirected) graph with each mixed graph, enabling the spectral analysis of the integrated adjacency matrix to connect the structural properties of the mixed graph and its associated graph. Furthermore, we define certain mixed graph structures and establish their relationships to the eigenvalues of the integrated adjacency matrix.

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arXiv2024

New matrices for spectral hypergraph theory, II

The properties of a hypergraph explored through the spectrum of its unified matrix was made by the authors in [26]. In this paper, we introduce three different hypergraph matrices: unified Laplacian matrix, unified signless Laplacian matrix, and unified normalized Laplacian matrix, all defined using the unified matrix. We show that these three matrices of a hypergraph are respectively identical to the Laplacian matrix, signless Laplacian matrix, and normalized Laplacian matrix of the associated graph. This allows us to use the spectra of these hypergraph matrices as a means to connect the structural properties of the hypergraph with those of the associated graph. Additionally, we introduce certain hypergraph structures and invariants during this process, and relate them to the eigenvalues of these three matrices.

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arXiv2024

New matrices for spectral hypergraph theory, I

We introduce a hypergraph matrix, named the unified matrix, and use it to represent the hypergraph as a graph. We show that the unified matrix of a hypergraph is identical to the adjacency matrix of the associated graph. This enables us to use the spectrum of the unified matrix of a hypergraph as a tool to connect the structural properties of the hypergraph with those of the associated graph. Additionally, we introduce certain hypergraph structures and invariants during this process, and relate them to the eigenvalues of the unified matrix.

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arXiv2022

Tensor join of hypergraphs and its spectra

In this paper, we introduce three operations on hypergraphs by using tensors. We show that these three formulations are equivalent and we commonly call them as the tensor join. We show that any hypergraph can be viewed as a tensor join of hypergraphs. Tensor join enable us to obtain several existing and new classes of operations on hypergraphs. We compute the adjacency, the Laplacian, the normalized Laplacian spectrum of weighted hypergraphs constructed by this tensor join. Also we deduce some results on the spectra of hypergraphs in the literature. As an application, we construct several pairs of the adjacency, the Laplacian, the normalized Laplacian cospectral hypergraphs by using the tensor join.

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arXiv2022

Automorphism group of the reduced power (di)graph of a finite group

We describe the full automorphism group of the directed reduced power graph and the undirected reduced power graph of a finite group. We compute the full automorphism groups of these graphs of several classes of finite groups. Also, we establish some relation between the automorphism group of the undirected reduced power graph (resp. the directed reduced power graph) and the automorphism group of the undirected power graph (resp. the directed power graph) of a finite group.

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arXiv2020

Spectra of generalized corona of graphs constrained by vertex subsets

In this paper, we introduce a generalization of corona of graphs. This construction generalizes the generalized corona of graphs (consequently, the corona of graphs), the cluster of graphs, the corona-vertex subdivision graph of graphs and the corona-edge subdivision graph of graphs. Further, it enables to get some more variants of corona of graphs as its particular cases. To determine the spectra of the adjacency, Laplacian and the signless Laplacian matrices of the above mentioned graphs, we define a notion namely, the coronal of a matrix constrained by an index set, which generalizes the coronal of a graph matrix. Then we prove several results pertain to the determination of this value. Then we determine the characteristic polynomials of the adjacency and the Laplacian matrices of this graph in terms of the characteristic polynomials of the adjacency and the Laplacian matrices of the constituent graphs and the coronal of some matrices related to the constituent graphs. Using these, we derive the characteristic polynomials of the adjacency and the Laplacian matrices of the above mentioned existing variants of corona of graphs, and some more variants of corona of graphs with some special constraints.

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arXiv2019

Spectra of $(H_1,H_2)$-merged subdivision graph of a graph

In this paper, we define a ternary graph operation which generalizes the construction of subdivision graph, $R-$graph, central graph. Also, it generalizes the construction of overlay graph (Marius Somodi \emph{et al.}, 2017), and consequently, $Q-$graph, total graph, and quasitotal graph. We denote this new graph by $[S(G)]^{H_1}_{H_2}$, where $G$ is a graph and, $H_1$ and $H_2$ are suitable graphs corresponding to $G$. Further, we define several new unary graph operations which becomes particular cases of this construction. We determine the Adjacency and Laplacian spectra of $[S(G)]^{H_1}_{H_2}$ for some classes of graphs $G$, $H_1$ and $H_2$. From these results, we derive the $L$-spectrum of the graphs obtained by the unary graph operations mentioned above. As applications, these results enable us to compute the number of spanning trees and Kirchhoff index of these graphs.

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arXiv2018

Some results on the reduced power graph of a group

The reduced power graph $\mathcal{RP}(G)$ of a group $G$ is the graph with vertex set $G$ and two vertices $u$ and $v$ are adjacent if and only if $\left\langle v\right\rangle \subset \left\langle u \right\rangle $ or $\left\langle u\right\rangle \subset \left\langle v \right\rangle $. The proper reduced power graph $\mathcal{RP}^*(G)$ of $G$ is the subgraph of $\mathcal{RP}(G)$ induced on $G\setminus \{e\}$. In this paper, we classify the finite groups whose reduced power graph (resp. proper reduced power graph) is one of complete $k$-partite, acyclic, triangle free or claw-free (resp. complete $k$-partite, acyclic, triangle free, claw-free, star or tree). In addition, we obtain the clique number and the chromatic number of $\mathcal{RP}(G)$ and $\mathcal{RP}^*(G)$ for any torsion group $G$. Also, for a finite group $G$, we determine the girth of $\mathcal{RP}^*(G)$. Further, we discuss the cut vertices, cut edges and perfectness of these graphs. Then we investigate the connectivity, the independence number and the Hamiltonicity of the reduced power graph (resp. proper reduced power graph) of some class of groups. Finally, we determine the number of components and the diameter of $\mathcal{RP}^*(G)$ for any finite group $G$.

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arXiv2016

Inclusion graph of subgroups of a group

For a finite group $G$, we define the inclusion graph of subgroups of $G$, denoted by $\mathcal I(G)$, is a graph having all the proper subgroups of $G$ as its vertices and two distinct vertices $H$ and $K$ in $\mathcal I(G)$ are adjacent if and only if either $H \subset K$ or $K \subset H$. In this paper, we classify the finite groups whose inclusion graph of subgroups is one of complete, bipartite, tree, star, path, cycle, disconnected, claw-free. Also we classify the finite abelian groups whose inclusion graph of subgroups is planar. For any given finite group, we estimate the clique number, chromatic number, girth of its inclusion graph of subgroups and for a finite abelian group, we estimate the diameter of its inclusion graph of subgroups. Among the other results, we show that some groups can be determined by their inclusion graph of subgroups

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arXiv2016

The complement of proper power graphs of finite groups

For a finite group $G$, the proper power graph $\mathscr{P}^*(G)$ of $G$ is the graph whose vertices are non-trivial elements of $G$ and two vertices $u$ and $v$ are adjacent if and only if $u \neq v$ and $u^m=v$ or $v^m=u$ for some positive integer $m$. In this paper, we consider the complement of $\mathscr{P}^*(G)$, denoted by ${\overline{\mathscr{P}^*(G)}}$. We classify all finite groups whose complement of proper power graphs is complete, bipartite, a path, a cycle, a star, claw-free, triangle-free, disconnected, planar, outer-planar, toroidal, or projective. Among the other results, we also determine the diameter and girth of the complement of proper power graphs of finite groups.

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arXiv2015

Coprime graph of subgroups of a group

Let $G$ be a group. We define the coprime graph of subgroups of $G$, denoted by $\mathcal P(G)$, is a graph whose vertex set is the set of all proper subgroups of $G$, and two distinct vertices are adjacent if and only if the order of the corresponding subgroups are coprime. In this paper, we study some connections between algebraic properties of a group and graph theoretic properties of its coprime graph.

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arXiv2015

Intersection graph of cyclic subgroups of groups

Let $G$ be a group. The intersection graph of cyclic subgroups of $G$, denoted by $\mathscr I_c(G)$, is a graph having all the proper cyclic subgroups of $G$ as its vertices and two distinct vertices in $\mathscr I_c(G)$ are adjacent if and only if their intersection is non-trivial. In this paper, we classify the finite groups whose intersection graph of cyclic subgroups is one of totally disconnected, complete, star, path, cycle. We show that for a given finite group $G$, $girth(\mathscr I_c (G)) \in \{3, \infty\}$. Moreover, we classify all finite non-cyclic abelian groups whose intersection graph of cyclic subgroups is planar. Also for any group $G$, we determine the independence number, clique cover number of $\mathscr I_c (G)$ and show that $\mathscr I_c (G)$ is weakly $\alpha$-perfect. Among the other results, we determine the values of $n$ for which $\mathscr I_c (\mathbb{Z}_n)$ is regular and estimate its domination number.

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arXiv2015

Toroidality and projective-planarity of intersection graphs of subgroups of finite groups

Let $G$ be a group. The intersection graph of subgroups of $G$, denoted by $\mathscr{I}(G)$, is a graph with all the proper subgroups of $G$ as its vertices and two distinct vertices in $\mathscr{I}(G)$ are adjacent if and only if the corresponding subgroups having a non-trivial intersection in $G$. In this paper, we classify the finite groups whose intersection graph of subgroups are toroidal or projective-planar. In addition, we classify the finite groups whose intersection graph of subgroups are one of bipartite, complete bipartite, tree, star graph, unicyclic, acyclic, cycle, path or totally disconnected. Also we classify the finite groups whose intersection graph of subgroups does not contain one of $K_5$, $K_4$, $C_5$, $C_4$, $P_4$, $P_3$, $P_2$, $K_{1,3}$, $K_{2,3}$ or $K_{1,4}$ as a subgraph. We estimate the girth of the intersection graph of subgroups of finite groups. Moreover, we characterize some finite groups by using their intersection graphs. Finally, we obtain the clique cover number of the intersection graph of subgroups of groups and show that intersection graph of subgroups of groups are weakly $\alpha$-perfect.

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arXiv2015

Classification of finite groups with toroidal or projective-planar permutability graphs

Let $G$ be a group. The permutability graph of subgroups of $G$, denoted by $\Gamma(G)$, is a graph having all the proper subgroups of $G$ as its vertices, and two subgroups are adjacent in $\Gamma(G)$ if and only if they permute. In this paper, we classify the finite groups whose permutability graphs are toroidal or projective-planar. In addition, we classify the finite groups whose permutability graph does not contain one of $K_{3,3}$, $K_{1,5}$, $C_6$, $P_5$, or $P_6$ as a subgraph.

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arXiv2015

Permutability graph of cyclic subgroups

Let $G$ be a group. \textit{The permutability graph of cyclic subgroups of $G$}, denoted by $\Gamma_c(G)$, is a graph with all the proper cyclic subgroups of $G$ as its vertices and two distinct vertices in $\Gamma_c(G)$ are adjacent if and only if the corresponding subgroups permute in $G$. In this paper, we classify the finite groups whose permutability graph of cyclic subgroups belongs to one of the following: bipartite, tree, star graph, triangle-free, complete bipartite, $P_n$, $C_n$, $K_4$, $K_{1,3}$-free, unicyclic. We classify abelian groups whose permutability graph of cyclic subgroups are planar. Also we investigate the connectedness, diameter, girth, totally disconnectedness, completeness and regularity of these graphs.

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arXiv2015

Permutability graphs of subgroups of some finite non-abelian groups

In this paper, we study the structure of the permutability graphs of subgroups, and the permutability graphs of non-normal subgroups of the following groups: the dihedral groups $D_n$, the generalized quaternion groups $Q_n$, the quasi-dihedral groups $QD_{2^n}$ and the modular groups $M_{p^n}$. Further, we investigate the number of edges, degrees of the vertices, independence number, dominating number, clique number, chromatic number, weakly perfectness, Eulerianness, Hamiltonicity of these graphs.

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