arXiv subjects

Mainak Basunia

Publications and source records attributed to Mainak Basunia.

4 recordsLinked to original sources

arXiv2026

Characterizing tricyclic graphs with pendant vertices having largest $A_{\alpha}$-spectral radius

For a graph $G$ with adjacency matrix $A(G)$ and degree diagonal matrix $D(G)$, the $A_{\alpha}$-matrix of $G$ is defined as \begin{equation*} A_{\alpha}(G) = \alpha D(G) + (1- \alpha) A(G), \text{ for any } \alpha \in [0,1]. \end{equation*} The $A_{\alpha}$-spectral radius of $G$ is the largest eigenvalue of the matrix $A_{\alpha}(G)$. A tricyclic graph of order $n$ is a simple connected graph with $n+2$ edges. In this paper, we characterize the unique graph having the largest $A_{\alpha}$-spectral radius for $\alpha \in [\frac{1}{2}, 1)$ among all tricyclic graphs of order $n$ with $k (\geq 1)$ pendant vertices. As an application, we derive a sufficient spectral condition (alternate to the edge condition) to guarantee the absence of the tricyclic structure in a graph with $k$ pendant vertices.

math.CO↗

arXiv2026

Exploring new upper and lower bounds for the $A_{\alpha}$-energy of graphs

Let $G$ be a graph on $n$ vertices and $m$ edges. For $\alpha \in [0,1]$, the $A_{\alpha}$-matrix of $G$ is defined as $A_{\alpha}(G) = \alpha D(G) + (1- \alpha) A(G)$, where $A(G)$ is the adjacency matrix and $D(G)$ is the degree diagonal matrix of $G$. If $\rho_1 \geq \rho_2 \ldots \geq \rho_n$ are the eigenvalues of $A_{\alpha}(G)$, the $A_{\alpha}$-energy of $G$ is defined as $E_{A_{\alpha}}(G) = \sum_{i=1}^{n} |\rho_i -\frac{2\alpha m}{n}|$. In this paper, we present novel upper and lower bounds for $E_{A_\alpha}(G)$ in terms of standard graph invariants, showing that each bound is sharp and identifying the specific graphs attaining them. For selected bounds, we provide brief comparative analysis with existing results, observing improved estimates. Furthermore, we establish new relations between $E_{A_\alpha}(G)$ and other well known graph energies, including adjacency, Laplacian, as well as the adjacency energy of the line graph.

math.CO↗

arXiv2026

$A_{\alpha}$-Spectra of $Q$- and $T$-Join Graphs with Applications to Cospectral Constructions

For $\alpha \in [0,1]$, the $A_{\alpha}$-matrix of a graph $G$ is defined by $A_{\alpha}(G) = \alpha D(G) + (1- \alpha) A(G)$, where $A(G)$ and $D(G)$ denote the adjacency matrix and the diagonal degree matrix of $G$, respectively. In this paper, we study the $A_{\alpha}$-characteristic polynomials and $A_{\alpha}$-spectra of graphs obtained via four recently introduced join operations, namely the $Q$-vertex join, $Q$-edge join, $T$-vertex join, and $T$-edge join, applied to two graphs $G_1$ and $G_2$. We derive explicit expressions for the $A_{\alpha}$-characteristic polynomials of these constructions when the first factor graph is regular. Furthermore, we determine the complete $A_{\alpha}$-spectra of these graphs in terms of the $A_{\alpha}$-spectra of the factor graphs, particularly when the second factor graph is regular or complete bipartite. The significance of these results lies in the fact that they enable efficient computation of the $A_{\alpha}$-spectra of large complex graphs arising from these joins, directly from the $A_{\alpha}$-spectra of the smaller constituent graphs, without explicitly constructing and handling the complex $A_{\alpha}$-matrices of those large graphs. Finally, as an application, we demonstrate how to construct infinitely many families of non-isomorphic graphs that are $A_{\alpha}$-cospectral.

math.CO↗

arXiv2020

On the $A_{\alpha}$-spectra of some join graphs

Let $G$ be a simple, connected graph and let $A(G)$ be the adjacency matrix of $G$. If $D(G)$ is the diagonal matrix of the vertex degrees of $G$, then for every real $\alpha \in [0,1]$, the matrix $A_{\alpha}(G)$ is defined as $$A_{\alpha}(G) = \alpha D(G) + (1- \alpha) A(G).$$ The eigenvalues of the matrix $A_{\alpha}(G)$ form the $A_{\alpha}$-spectrum of $G$. Let $G_1 \dot{\vee} G_2$, $G_1 \underline{\vee} G_2$, $G_1 \langle \textrm{v} \rangle G_2$ and $G_1 \langle \textrm{e} \rangle G_2$ denote the subdivision-vertex join, subdivision-edge join, $R$-vertex join and $R$-edge join of two graphs $G_1$ and $G_2$, respectively. In this paper, we compute the $A_{\alpha}$-spectra of $G_1 \dot{\vee} G_2$, $G_1 \underline{\vee} G_2$, $G_1 \langle \textrm{v} \rangle G_2$ and $G_1 \langle \textrm{e} \rangle G_2$ for a regular graph $G_1$ and an arbitrary graph $G_2$ in terms of their $A_{\alpha}$-eigenvalues. As an application of these results, we construct infinitely many pairs of $A_{\alpha}$-cospectral graphs.

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