cospectral graphs

In this paper, we define duplication corona, duplication neighborhood corona and duplication edge corona of two graphs. We compute their adjacency spectrum, Laplacian spectrum and signless Laplacian. As an application, our results enable us to construct infinitely many pairs of cospectral graphs and also integral graphs.

Let $n$ be any positive integer, the friendship graph $F_n$ consists of $n$ edge-disjoint triangles that all of them meeting in one vertex. A graph $G$ is called cospectral with a graph $H$ if their adjacency matrices have the same eigenvalues. Recently in href{http://arxiv.org/pdf/1310.6529v1.pdf}{http://arxiv.org/pdf/1310.6529v1.pdf} it is proved that if $G$ is any graph cospectral with $F_n$ ($nneq 16$), then $Gcong F_n$. In this note, we give a proof of a special case of the latter: Any connected graph cospectral with $F_n$ is isomorphic to $F_n$.Our proof is independent of ones given in href{http://arxiv.org/pdf/1310.6529v1.pdf http://arxiv.org/pdf/1310.6529v1.pdf} and the proofs are based on our recent results iven in [{em Trans. Comb.}, {bf 2} no. 4 (2013) 37-52.] using an upper bound for the argest eigenvalue of a connected graph given in {em J. Combinatorial Theory Ser. B} {bf 81} (2001) 177-183.].

Let $n$ be any positive integer and let $F_n$ be the friendship (or Dutch windmill) graph with $2n+1$ vertices and $3n$ edges. Here we study graphs with the same adjacency spectrum as the $F_n$. Two graphs are called cospectral if the eigenvalues multiset of their adjacency matrices are the same. Let $G$ be a graph cospectral with $F_n$. Here we prove that if $G$ has no cycle of length $4$ or $5$, then $Gcong F_n$. Moreover if $G$ is connected and planar then $Gcong F_n$.All but one of connected components of $G$ are isomorphic to $K_2$.The complement $overline{F_n}$ of the friendship graph is determined by its adjacency eigenvalues, that is, if $overline{F_n}$ is cospectral with a graph $H$, then $Hcong overline{F_n}$.