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UDC 517.9Two graphs are said to be $Q$-cospectral if they have the same signless Laplacian spectrum.A graph is said to be DQS if there are no other nonisomorphic graphs $Q$-cospectral with it. A tree is called double starlike if it has exactly two vertices of degree greater than 2.Let $H_n(p,q)$ with $n \ge 2,$ $p \geq q \geq 2$ denote the double starlike tree obtained by attaching $p$ pendant vertices to one pendant vertex of the path $P_n$ and $q$ pendant vertices to the other pendant vertex of $P_n.$ In this paper, we prove that $H_n(p,q)$ is DQS for $n\ge 2,$ $p\geq q\geq 2.$

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.

A peacock graph $$PG=(i, j, k\,; b_1, b_2,\ldots ,b_s)$$, where $$i,j,k\ge 3$$ is a graph consisting of three cycles $$C_i$$, $$C_j$$, $$C_k$$ and $$s\,(\ge 1)$$ paths $$P_{b_1+1}, P_{b_2+1},\ldots , P_{b_s+1}$$ intersecting in a single vertex that all meet in one vertex. A graph G is said to be determined by the spectrum of its Laplacian matrix (DLS, for short) if every graph with the same Laplacian spectrum is isomorphic to G. If $$s=1$$, then the peacock graph is called clover graph. In Wang and Wang (Linear Multilinear Algebra 63(12):2396–2405, 2015), it was proved that all clover graphs are DLS. In this paper, we generalize this result and show that all peacock graphs are DLS.

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, Comment: Accepted in AKCE International Journal of Graphs and Combinatorics(To appear). arXiv preprint arXiv:1803.06135 has been accepted in Carpathian Mathematical Publications. arXiv admin note: text overlap with arXiv:1803.06135

A multicone graph is defined to be the join of a clique and a regular graph. Let w , n and m be natural numbers, and let K w and K n , n denote a complete graph and a complete bipartite graph, respectively. In this work, it is proved that connected multicone graphs K w ▽ m K n , n , natural generalizations of friendship graphs, are determined by their adjacency spectra as well as their Laplacian spectra. Also, we show that the complement of multicone graphs K w ▽ K n , n is determined by their adjacency spectra. Furthermore, we prove that any graph cospectral with a multicone graph K w ▽ m K n , n is perfect with respect to its adjacency (Laplacian) spectra. At the end of the paper, we pose two problems for further research.

A multicone graph is defined to be the join of a clique and a regular graph. Let Km and denote a complete graph on m vertices and the complement of a complete graph on n vertices, respectively. In ...

Finding and discovering any class of graphs which are determined by their spectra is always an important and interesting problem in the spectral graph theory. The main aim of this study is to characterize two classes of multicone graphs which are determined by both their adjacency and Laplacian spectra. A multicone graph is defined to be the join of a clique and a regular graph. Let Kw denote a complete graph on w vertices, and let m be a positive integer number. In A.Z.Abdian (2016) it has been shown that multicone graphs Kw ▽ P17 and Kw ▽ S are determined by both their adjacency and Laplacian spectra, where P17 and S denote the Paley graph of order 17 and the Schlafli graph, respectively. In this paper, we generalize these results and we prove that multicone graphs Kw ▽mP17 and Kw▽mS are determined by their adjacency spectra as well as their Laplacian spectra.

A graph G is said to be determined by its Laplacian spectrum (DLS) if every graph with the same Laplacian spectrum is isomorphic to G. A graph which is a collection of hexagons (lengths of these cycles can be different) all sharing precisely one vertex is called a spinner graph. A tree with exactly one vertex of degree greater than 2 is called a starlike tree. If a spinner graph and a starlike tree are joined by merging their vertices of degree greater than 2, then the resulting graph is called a tarantula graph. It is known that spinner graphs and starlike trees are DLS. In this paper, we prove that tarantula graphs are determined by their Laplacian spectrum.

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 $K_w\bigtriangledown mECP_{l}^{k}$ is determined by its adjacency spectra as well as its Laplacian spectra, where $ ECP_{l}^{k}={K_{\underbrace {{3^k},\,{3^k},\,...,\,{3^k}}_{l\,times}}}$. 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.

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.