# Spectra of $(H_1,H_2)$-merged subdivision graph of a graph

## TL;DR

This paper introduces a new ternary graph operation that generalizes several existing graph constructions, and determines the spectra and structural properties of the resulting graphs for various classes.

## Contribution

It defines a novel graph operation unifying multiple known constructions and derives spectral and combinatorial properties for these generalized graphs.

## Key findings

- Derived adjacency and Laplacian spectra for the new graph operation.
- Computed the number of spanning trees and Kirchhoff index for the generalized graphs.
- Unified several graph operations under a single framework.

## Abstract

In this paper, we define a ternary graph operation which generalizes the construction of subdivision graph, $R-$graph, central graph. Also, it generalizes the construction of overlay graph (Marius Somodi \emph{et al.}, 2017), and consequently, $Q-$graph, total graph, and quasitotal graph. We denote this new graph by $[S(G)]^{H_1}_{H_2}$, where $G$ is a graph and, $H_1$ and $H_2$ are suitable graphs corresponding to $G$. Further, we define several new unary graph operations which becomes particular cases of this construction. We determine the Adjacency and Laplacian spectra of $[S(G)]^{H_1}_{H_2}$ for some classes of graphs $G$, $H_1$ and $H_2$. From these results, we derive the $L$-spectrum of the graphs obtained by the unary graph operations mentioned above. As applications, these results enable us to compute the number of spanning trees and Kirchhoff index of these graphs.