# Laplacian Spectra of Regular Graph Transformations

## TL;DR

This paper introduces a new class of graph transformations based on parameters and derives the Laplacian spectra of these transformed graphs for regular graphs, linking them to the original spectra and graph parameters.

## Contribution

It provides a complete description of the Laplacian spectra of (x,y,z)-transformations of regular graphs, connecting the spectra to original graph properties.

## Key findings

- Laplacian polynomial of G(x,y,z) depends on |V|, r, and G's Laplacian spectrum.
- Explicit formulas for spectra of transformed graphs.
- Applicable to regular graphs with various parameter choices.

## Abstract

For any given graph G = (V,E) we define in a certain way a new graph G(x,y,z) with the vertex set V\cup E depending on parameters x,y,z from {0,1, +, -} and call graph G(x,y,z) the (x,y,z)-transformation of G. It turns out that if G is an r-regular graph, then the Laplacian polynomial of G(x,y,z) is a function of |V|, r, and the Laplacian spectrum of G. We give a complete description of this function.