The generalized adjacency spectra of triple vertex and triple edge join of graphs

Articles in Press

Document Type : Research Paper

Authors

Department of Applied Sciences, Tezpur University, Napaam-784028, Assam, India.

10.22034/as.2026.23908.1849

Abstract

Consider three graphs $G_1$, $G_2$, and $H$ with orders $n_1,~n_2, n_3,$ and sizes $m_1,m_2,m_3,$ respectively, where the sets of vertices for all three graphs are disjoint. Let $S(G_1, H)$ be the graph obtained from $G_1$ and $H$ in the following way: 
(1) Delete all the edges of $G_1$ and consider $m_1$ disjoint copies of $H$. 
(2) Join each vertex of the $i^{th}$ copy of $H$ to the end vertices of the $i^{th}$ edge of $G_1$.
Using this construction, the graph $G_1(\vee_H)G_2$ is defined to be the graph obtained from $S(G_1, H)$ by joining every vertex of $G_1$ to every vertex of $G_2$ (see \cite{som1}). In a similar manner, we define $G_1(\Lambda_H)G_2$ to be the graph formed by connecting every vertex in each of the $m_1$ inserted copies of $H$ in $S(G_1, H)$ to all vertices of $G_2$. We refer to $G_1(\vee_H)G_2$ and $G_1(\Lambda_H)G_2$ as the \emph{triple vertex join} and \emph{triple edge join} of $G_1$ and $G_2$ with respect to $H$, respectively.
In this article, we determine the generalized adjacency spectra (also known as the $A_\alpha$-spectra) of $G_1(\vee_H)G_2$ when $G_1$ is regular, and of $G_1(\Lambda_H)G_2$ when both $G_1$ and $H$ are regular. Furthermore, we present an application demonstrating the construction of infinitely many $A_\alpha$-cospectral graphs.

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References

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Algebraic Structures and Their Applications

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Available Online from 16 September 2026

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