On the $A_{\alpha}$-spectrum of the $k$-splitting signed graph and neighbourhood coronas

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Document Type : Original paper

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Department of Mathematics, University of Kashmir, Srinagar, India

Abstract

Let $\Sigma=(G,\sigma)$ be a signed graph with adjacency matrix $A(\Sigma)$ and $D(G)$ be the diagonal matrix of its vertex degrees. For any real $\alpha\in [0,1]$, the $A_{\alpha}$-matrix of a signed graph $\Sigma$ is defined as $A_{\alpha}(\Sigma)=\alpha D(G)+(1-\alpha)A(\Sigma)$. Given a signed graph $\Sigma$ with vertex set $V=\{v_1, v_2,\dots, v_n\}$, the $k$-splitting signed graph $SP_k(\Sigma)$ of $\Sigma$ is obtained by adding to each vertex $v\in V(\Sigma)$ new $k$ vertices say $u^1, u^2, \ldots, u^k$ and joining every neighbour say $u$ of the vertex $v$ to $u^i$, $1\le i\le k$ by an edge which inherits the sign from $uv$. In this paper, we determine the $A_{\alpha}$-spectrum of $SP_k(\Sigma)$ in case of $\Sigma$ being a regular signed graph. For $k=1$, we introduce two distinct coronas of signed graphs $\Sigma_1$ and $\Sigma_2$ based on $SP_1(\Sigma_1)$, namely the splitting V-vertex neighbourhood corona and the splitting S-vertex neighbourhood corona. By examining the $A_{\alpha}$-characteristic polynomial of the resulting signed graphs, we derive their $A_{\alpha}$-spectra under certain regularity conditions on the constituent signed graphs. As applications, we use these results to construct infinite pairs of nonregular $A_{\alpha}$-cospectral signed graphs.

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References

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Communications in Combinatorics and Optimization

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Available Online from 25 June 2024

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