The Kite graph, denoted by Kite p,q is obtained by appending a complete graph K p to a pendant vertex of a path P q . In this paper, firstly we show that no two non-isomorphic kite graphs are cospectral w.r.t adjacency matrix. Let G be a graph which is cospectral with Kite p,q and the clique number of G is denoted by w(G). Then, it is shown that w(G) ≥ p − 2q + 1. Also, we prove that Kite p,2 graphs are determined by their adjacency spectrum.

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The pineapple graph K q p is obtained by appending q pendant edges to a vertex of a complete graph K p (q ≥ 1, p ≥ 3). Zhang and Zhang (2009) [7] claim that the pineapple graphs are determined by their adjacency spectrum. We show that their claim is false by constructing graphs which are cospectral and non-isomorphic with K q p for every p ≥ 4 and various values of q. In addition we prove that the claim is true if q = 2, and refer to the literature for q = 1, p = 3, and (p, q) = (4, 3).

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A mixed extension of a graph G is a graph H obtained from G by replacing each vertex of G by a clique or a coclique, whilst two vertices in H corresponding to distinct vertices x and y of G are adjacent whenever x and y are adjacent in G. If G is the path P 3 , then H has at most three adjacency eigenvalues unequal to 0 and −1. Recently, the first author classified the graphs with the mentioned eigenvalue property. Using this classification we investigate mixed extension of P 3 on being determined by the adjacency spectrum. We present several cospectral families, and with the help of a computer we find all graphs on at most 25 vertices that are cospectral with a mixed extension of P 3 . present several infinite families of cospectral graphs with all but three eigenvalues equal to 0 or −1.2 Mixed extensions of P 3 Let G be a graph with vertex set V (G) = {1, . . . , m} and let V 1 , . . . , V m be mutually disjoint nonempty finite sets. A graph H with vertex set V (H) = V 1 ∪ . . . ∪V m is defined as follows. For each i ∈ {1, . . . , m}, all vertices of V i are either mutually adjacent (form a clique), or mutually nonadjacent (form a coclique). For any u ∈ V i and v ∈ V j (i = j) {u, v} in an edge in H if and only if {i, j} is an edge in G. The graph H is called a mixed extension of G. A mixed extension is represented by an m-tuple (t 1 , . . . , t m ) of nonzero integers, where t i > 0 indicates that V i is a clique of order t i and t i < 0 means that V i is a coclique of order −t i . A mixed extension of G is a special case of a generalized composition or G-join, introduced in [3] and [6], respectively. We refer to [5], [3] and [6] for basic results on mixed extensions, and to [1] or [4] for graph spectra.Suppose H is a mixed extension of the path P 3 of type (t 1 , t 2 , t 3 ). Then the adjacency matrix of H admits the following structure. (As usual, J is an all-ones matrix, J n is the n × n all-ones matrix, and I n is the identity matrix of order n.)