arXiv subjects

Najiya V K

Publications and source records attributed to Najiya V K.

5 recordsLinked to original sources

arXiv2024

Constructions of $A_α$-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 $A_α$-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 $A_α$-cospectral graphs.

math.CO↗

arXiv2024

$A_α$-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 $α\in [0, 1]$, the $A_α$-matrix is defined as $A_α(G) = αD(G) + (1 - α)A(G)$. In this paper, we compute the $A_α$-characteristic polynomial, $A_α$-spectra and $A_α$-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 $A_α$-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 $A_α$-equieneregetic and $A_α$-borderenergetic.

math.CO↗

arXiv2024

Construction of non-regular $A_α$-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 $A_α$-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 $A_α$-spectrum of these graphs. As an application, we construct non-regular, non-isomorphic graphs that are $A_α$-cospectral.

math.CO↗

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 $A_α$-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 $A_α$-characteristic polynomial of $k$-coalescence of two graphs and then compute the $A_α$-spectra of $k$-coalescence of two complete graphs. In addition, we estimate the $A_α$-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}}).

math.CO↗

arXiv2023

A study on $A_α$-spectrum and $A_α$-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 $A_α$-matrix of a graph $G$ is defined as $A_α(G) = αD(G) + (1-α)A(G)$, $α\in [0, 1]$, 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 $A_α$-eigenvalues for unitary addition Cayley graph and its complement. We determine bounds for $A_α$-eigenvalues of unitary addition Cayley graph when its order is odd. Consequently, we compute the $A_α$-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 $A_α$-borderenergetic and $A_α$-hyperenergetic graphs and observe some classes for each.

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