A graph G is said to be determined by its generalized spectrum (DGS for short) if for any graph H, H and G are cospectral with cospectral complements implies that H is isomorphic to G. Let Ĝ be the graph obtained from G by adding a pendent edge at every vertex of G. We show that Ĝ is DGS if and only if G is DGS for some graph G. This gives a simple way to construct large DGS graphs from small ones explicitly. In particular, we show that every graph in the infinite sequence G, Ĝ , Ĝ , · · · is DGS, for some DGS graph G.

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A graph G is said to be determined by its spectrum if any graph having the same spectrum as G is isomorphic to G. Let K n \ P ℓ be the graph obtained from K n by removing edges of P ℓ , where P ℓ is a path of length ℓ − 1 which is a subgraph of a complete graph K n . Cámara and Haemers [11] conjectured that K n \P ℓ is determined by its adjacency spectrum for every 2 ≤ ℓ ≤ n. In this paper we show that the conjecture is true for 7 ≤ ℓ ≤ 9.

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