In the past decades, graphs that are determined by their spectrum have received more attention, since they have been applied to several fields, such as randomized algorithms, combinatorial optimization problems and machine learning. An important part of spectral graph theory is devoted to determining whether given graphs or classes of graphs are determined by their spectra or not. So, finding and introducing any class of graphs which are determined by their spectra can be an interesting and important problem. A graph is said to be DQS if there is no other non-isomorphic graph with the same signless Laplacian spectrum. For a DQS graph G, we show that G ∪ rK1 ∪ sK2 is DQS under certain conditions, where r, s are natural numbers and K1 and K2 denote the complete graphs on one vertex and two vertices, respectively. Applying these results, some DQS graphs with independent edges and isolated vertices are obtained.
In this study we investigate the spectra of the family of connected multicone graphs. A multicone graph is defined to be the join of a clique and a regular graph. Let r, t and s be natural numbers, and let Kr denote a complete graph on r vertices. It is proved that connected multicone graphs Kr ▽ sKt, a natural generalization of friendship graphs, are determined by their adjacency spectra as well as their Laplacian spectra. Also, we show that the complement of multicone graphs Kr ▽ sKt are determined by their adjacency spectra, where s = 2.
Abstract. This paper deals with graphs that are known as multicone graphs. A multicone graph is a graph obtained from the join of a clique and a regular graph. Let w, l, m be natural numbers and k is a natural number. It is proved that any connected graph cospectral with multicone graph Kw mECP k l is determined by its adjacency spectra as well as its Laplacian spectra, where. Also, we show that complements of some of these multicone graphs are determined by their adjacency spectra. Moreover, we prove that any connected graph cospectral with these multicone graphs must be perfect. Finally, we pose two problems for further researches.Mathematics Subject Classification (2010): 05C50.
In the past decades, graphs that are determined by their spectrum have received much more and more attention, since they have been applied to several fields, such as randomized algorithms, combinatorial optimization problems and machine learning. An important part of spectral graph theory is devoted to determining whether given graphs or classes of graphs are determined by their spectra or not. So, finding and introducing any class of graphs which are determined by their spectra can be an interesting and important problem. The main aim of this study is to characterize two classes of multicone graphs which are determined by their adjacency, Laplacian and signless Laplacian spectra. A multicone graph is defined to be the join of a clique and a regular graph. Let [Formula: see text] denote a complete graph on [Formula: see text] vertices. In the paper, we show that multicone graphs [Formula: see text] and [Formula: see text] are determined by both their adjacency spectra and their Laplacian spectra, where [Formula: see text] and [Formula: see text] denote the Local Higman–Sims graph and the Local [Formula: see text] graph, respectively. In addition, we prove that these multicone graphs are determined by their signless Laplacian spectra.
A graph G is said to be determined by the spectrum of its Laplacian matrix (DLS) if every graph with the same spectrum is isomorphic to G. In some recent papers it is proved that the friendship graphs and starlike trees are DLS. If a friendship graph and a starlike tree are joined by merging their vertices of degree greater than two, then the resulting graph is called a path-friendship graph. In this paper, it is proved that the path-friendship graphs, a natural generalization of friendship graphs and starlike trees, are also DLS. Consequently, using these results we provide a solution for an open problem.
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