# Monster graphs are determined by their Laplacian spectra

## DOI:

https://doi.org/10.33044/revuma.1769## Abstract

A graph $G$ is determined by its Laplacian spectrum (DLS) if every graph with the same Laplacian spectrum is isomorphic to $G$. A multi-fan graph is a graph of the form $(P_{n_1}\cup P_{n_2}\cup \cdots \cup P_{n_k})\bigtriangledown K_1$, where $K_1$ denotes the complete graph of size 1, $P_{n_1}\cup P_{n_2}\cup \cdots \cup P_{n_k}$ is the disjoint union of paths $P_{n_i}$, $n_i\geq 1$ and $1 \leq i \leq k$; and a starlike tree is a tree with exactly one vertex of degree greater than 2. If a multi-fan graph and a starlike tree are joined by identifying their vertices of degree more than 2, then the resulting graph is called a monster graph. In some earlier works, it was shown that all multi-fan and path-friendship graphs are DLS. The aim of this paper is to generalize these facts by proving that all monster graphs are DLS.

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Copyright (c) 2022 Ali Z. Abdian, Ali R. Ashrafi, Lowell W. Beineke, Mohammad Reza Oboudi, Gholam Hossein Fath-Tabar

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