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Chithra A V

Publications and source records attributed to Chithra A V.

16 recordsLinked to original sources
arXiv2026

Eccentricity energy change of coalescence of graphs due to edge deletion

The eccentricity matrix of a graph is obtained from the distance matrix by keeping the largest entries in their row or column, and the remaining entries are replaced by zeros. The eccentricity energy of a graph is the sum of the absolute values of the eigenvalues of its eccentricity matrix. In this paper, we investigate the effect of edge deletion on the eccentricity energy of graphs of the form $$G=K_{2n}\circ_{n} K_{2n}\circ_{n}\cdots \circ_{n} K_{2n}, (\text{\textit{l} copies of } K_{2n}),$$ where $n\geq 3,$ $l\geq 2,$ and $\circ_{n}$ denotes the $n-$coalescence of graphs, and prove that the eccentricity energy increases whenever an edge is removed. This result identifies a class of graphs whose eccentricity energy exhibits monotonic growth under edge deletion.

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arXiv2025

Eccentricity spectral properties of $\mathcal{C}$-graphs

A cograph is a simple graph that contains no induced path on four vertices. In this paper, we consider $\mathcal{C}$-graphs, which are a specific class of cographs, defined as $$\overline{\overline{\overline{K_{\alpha_{1}}}\cup K_{\alpha_{2}}}\cup \cdots \cup K_{\alpha_{2k}}},$$ %\text{ where } k \geq 2, \alpha_{2k}\geq 2,$$ where $k \geq 2$, $\alpha_{2k} \geq 2$, and $K_{\alpha_{i}}$ denotes the complete graph on $\alpha_{i}$ vertices. We investigate the spectral properties of the eccentricity matrix of this particular class of cographs. Additionally, we determine the irreducibility and inertia of the eccentricity matrix of $\mathcal{C}$-graphs. Furthermore, we identify an interval $(-1-\sqrt{2},-2)\cup (-2,0)$ in which these graphs have no eccentricity eigenvalues.

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arXiv2024

Eccentricity spectrum of join of central graphs and Eccentricity Wiener index of graphs

The eccentricity matrix of a simple connected graph is derived from its distance matrix by preserving the largest non-zero distance in each row and column, while the other entries are set to zero. This article examines the $\epsilon$-spectrum, $\epsilon$-energy, $\epsilon$-inertia and irreducibility of the central graph (respectively complement of the central graph) of a triangle-free regular graph(respectively regular graph). Also look into the $\epsilon-$spectrum and the irreducibility of different central graph operations, such as central vertex join, central edge join, and central vertex-edge join. We also examine the $\epsilon-$ energy of some specific graphs. These findings allow us to construct new families of $\epsilon$-cospectral graphs and non $\epsilon$-cospectral $\epsilon-$equienergetic graphs. Additionally, we investigate certain upper and lower bounds for the eccentricity Wiener index of graphs. Also, provide an upper bound for the eccentricity energy of a self-centered graph.

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arXiv2024

Metric dimension and Zagreb indices of essential ideal graph of a finite commutative ring

Let $R$ be a commutative ring with unity. The essential ideal graph $\mathcal{E}_{R}$ of $R$ is a graph whose vertex set consists of all nonzero proper ideals of \textit{R}. Two vertices $\hat{I}$ and $\hat{J}$ are adjacent if and only if $\hat{I}+ \hat{J}$ is an essential ideal. In this paper, we characterize the graph $\mathcal{E}_{R}$ as having a finite metric dimension. Additionally, we identify that the essential ideal graph and annihilating ideal graph of the ring $\mathbb{Z}_{n}$ are isomorphic whenever $n$ is a product of distinct primes. Also, we estimate the metric dimension of the essential ideal graph of the ring $\mathbb{Z}_{n}$. Furthermore, we determine the topological indices, namely the first and the second Zagreb indices, of $\mathcal{E}_{\mathbb Z_n}$.

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arXiv2024

Constructions of $A_\alpha$-cospectral graphs using some corona operations

Let $ G_1 \circledast G_2$,$ G_1 \sqcupdot G_2 $ and $ G_1 \sqcupplus G_2$ denote the total corona, $Q$-vertex corona and $Q$-edge corona of two graphs $ G_1$ and $ G_2 $, respectively. In this paper, we compute the $A_\alpha$-spectrum of $ G_1 \circledast G_2$,$ G_1 \sqcupdot G_2 $ and $ G_1 \sqcupplus G_2$ for regular graphs $ G_1$ and $ G_2$. As an application, we construct infinitely many pairs of $A_\alpha$-cospectral graphs.

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arXiv2024

On the resistance regular graphs

For a connected graph $G$, its resistance distance matrix is denoted by $R(G)$. A graph is called resistance regular if all the row (or column) sums of $R(G)$ are equal. We provide a necessary and sufficient condition for a simple connected graph to be resistance regular. Additionally, we establish sharp bounds for the resistance spectral radius and present various bounds for the resistance energy of $G$. Furthermore, we compute the resistance spectrum and resistance energy of some resistance regular graphs.

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arXiv2024

On irreducibility of eccentricity matrix of graphs and construction of $\epsilon-$equienergetic graphs

The eccentricity matrix $\epsilon(G)$, of a connected graph $G$ is obtained by retaining the maximum distance from each row and column of the distance matrix of $G$ and the other entries are assigned with 0. In this paper, we discuss the eccentricity spectrum of subdivision vertex (edge) join of regular graphs. Also, we obtain new families of graphs having irreducible or reducible eccentricity matrix. Furthermore, we use these results to construct infinitely many $\epsilon-$cospectral graph pairs as well as infinitely many pairs and triplets of $\epsilon-$cospectral $\epsilon-$equienergetic graphs. Moreover, we present some new family of $\epsilon-$integral graphs.

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arXiv2024

Resistance distance and Kirchhoff index in central vertex join and central edge join of two graphs

The central graph $C(G)$ of a graph $G$ is the graph obtained by inserting a new vertex into each edge of $G$ exactly once and joining all the non-adjacent vertices in $G$. Let $G_1$ and $G_2$ be two vertex disjoint graphs. The central vertex join of $G_1$ and $G_2$ is the graph $ G_1\dot{\vee} G_2$, is obtained from $C(G_1)$ and $G_2$ by joining each vertex of $G_1$ with every vertex of $G_2$. The central edge join of $G_1$ and $G_2$ is the graph $ G_1\veebar G_2$, is obtained from $C(G_1)$ and $G_2$ by joining each vertex corresponding to the edges of $G_1$ with every vertex of $G_2$. In this article, we obtain formulae for the resistance distance and Kirchhoff index of $G_1\dot{\vee} G_2$ and $ G_1\veebar G_2$. In addition, we provide the resistance distance, Kirchhoff index, and Kemeny's constant of the central graph of a graph.

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arXiv2024

$A_\alpha$-energy of graphs formed by some unary operations

Let $G $ be a graph on $p$ vertices with adjacency matrix $A(G)$ and degree matrix $D(G)$. For each $\alpha \in [0, 1]$, the $A_\alpha$-matrix is defined as $A_\alpha (G) = \alpha D(G) + (1 - \alpha)A(G)$. In this paper, we compute the $A_\alpha$-characteristic polynomial, $A_\alpha$-spectra and $A_\alpha$-energy of some non-regular graphs obtained from unary operations on graphs like middle graph, central graph, m-splitting, and closed splitting graph. Also, we determine the $A_\alpha$-energy of regular graphs like m-shadow, closed shadow, extended bipartite double graph, iterated line graph and m-duplicate graph. Furthermore, we identified some graphs that are $A_\alpha$-equieneregetic and $A_\alpha$-borderenergetic.

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arXiv2023

On the Adjacency and Seidel Spectra of Hypergraphs

A hypergraph generalizes the concept of an ordinary graph. In an ordinary graph, edges connect pairs of vertices, whereas in a hypergraph, hyperedges can connect multiple vertices at a time. In this paper, we obtain a relationship between the characteristic polynomial of Seidel and adjacency matrices of hypergraph and also compute all the eigenvalues of some k-uniform hypergraphs. Moreover, we estimate the adjacency and Seidel spectra of the uniform double hyperstar and sunflower hypergraph. In addition to that, we determine the Seidel spectrum and main Seidel eigenvalues of hyperstar.

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arXiv2023

Resistance distance in $k$-coalescence of certain graphs

Any graph can be considered as a network of resistors, each of which has a resistance of $1 \Omega.$ The resistance distance $r_{ij}$ between a pair of vertices $i$ and $j$ in a graph is defined as the effective resistance between $i$ and $j$. This article deals with the resistance distance in the $k$-coalescence of complete graphs. We also present its results in connection with the Kemeny's constant, Kirchhoff index, additive degree-Kirchhoff index, multiplicative degree-Kirchhoff index and mixed degree-Kirchhoff index. Moreover, we obtain the resistance distance in the $k$-coalescence of a complete graph with particular graphs. As an application, we provide the resistance distance of certain graphs such as the vertex coalescence of a complete bipartite graph with a complete graph, a complete bipartite graph with a star graph, the windmill graph, pineapple graph, etc.

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arXiv2023

A study on $A_\alpha$-spectrum and $A_\alpha$-energy of unitary addition Cayley graphs

The unitary addition Cayley graph $G_n$, $n\in Z^+$ is the graph whose vertex set is $Z_n$, the ring of integers modulo $n$ and two vertices $u$ and $v$ are adjacent if and only if $u + v \in \cup_n$ where $\cup_n$ is the set of all units of the ring. The $A_\alpha$-matrix of a graph $G$ is defined as $A_\alpha (G) = \alpha D(G) + (1-\alpha)A(G)$, $\alpha \in [0, 1]$, where $D(G)$ is the diagonal matrix of vertex degrees and $A(G)$ is the adjacency matrix of $G$. In this paper, we investigate the $A_\alpha$-eigenvalues for unitary addition Cayley graph and its complement. We determine bounds for $A_\alpha$-eigenvalues of unitary addition Cayley graph when its order is odd. Consequently, we compute the $A_\alpha$-energy of both $G_n$ and its complement, $\overline{G_n}$, for $n={p}^m$ where ${p}$ is a prime number and $n$ even. Moreover, we obtain some bounds for energies of $G_n$ and $\overline{G}_n$ when $n$ is odd. We also define $A_\alpha$-borderenergetic and $A_\alpha$-hyperenergetic graphs and observe some classes for each.

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arXiv2023

A study on $k$-coalescence of two graphs

The $k$-coalescence of two graphs is obtained by merging a $k$-clique of each graph. The $A_\alpha$-matrix of a graph is the convex combination of its degree matrix and adjacency matrix. In this paper, we present some structural properties of a non-regular graph which is obtained from the $k$-coalescence of two graphs. Also, we derive the $A_\alpha$-characteristic polynomial of $k$-coalescence of two graphs and then compute the $A_\alpha$-spectra of $k$-coalescence of two complete graphs. In addition, we estimate the $A_\alpha$-energy of $k$-coalescence of two complete graphs. Furthermore, we obtain some topological indices of vertex coalescence of two graphs, and as an application, we determine some indices of some family of graphs. From these results, we calculate the Wiener index, hyper-Wiener index etc. of the organic compound 1,2-dicyclohexylethane(\ce{C_{14}H_{26}}).

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arXiv2022

Construction of non-regular $A_\alpha$-cospectral graphs from some join of graphs

Cospectral graphs are a fascinating concept in graph theory, where two non-isomorphic graphs possess identical sets of eigenvalues. In this paper, we compute the $A_\alpha$-characteristic polynomial of neighbour and non-neighbour splitting join, neighbour and non-neighbour shadow join, central vertex and edge join and duplicate join of two graphs. In addition, when $\graphene_1$ and $\graphene_2$ are regular, we compute the $A_\alpha$-spectrum of these graphs. As an application, we construct non-regular, non-isomorphic graphs that are $A_\alpha$-cospectral.

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arXiv2021

Energy and Randic' energy of special graphs

In this paper, we determine the Randic' energy of the m-splitting graph, the m-shadow graph and the m-duplicate graph of a given graph, m being an arbitrary integer. Our results allow the construction of an infinite sequence of graphs having the same Randic' energy. Further, we determine some graph invariants like the degree Kirchhoff index, the Kemeny's constant and the number of spanning trees of some special graphs. From our results, we indicate how to obtain infinitely many pairs of equienergetic graphs, Randic' equienergetic graphs and also, infinite families of integral graphs.

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