The Lollipop Graph is Determined by its Spectrum

An even (resp. odd) lollipop is the coalescence of a cycle of even (resp. odd) length and a path with pendant vertex as distinguished vertex. It is known that the odd lollipop is determined by its spectrum and the question is asked by W. Haemers, X. Liu and Y. Zhang for the even lollipop. We revisit the proof for odd lollipop, generalize it for even lollipop and therefore answer the question. Our proof is essentially based on a method of counting closed walks. Let G be a simple graph with n vertices and A its adjacency matrix, Q G (X) denotes its characteristic polynomial and λ 1 (G) ≥ λ 2 (G) ≥ · · · ≥ λ n (G) the associated eigenvalues; λ 1 (G) is the spectral radius of G. It is known that some informations about the graph structure can be deduced from these eigenvalues such as the number of edges or the length of the shortest odd cycle; but the reverse question Which graphs are determined by their spectrum ? (asked, among others, in [4]) is far from being solved; some partial results exist [5, 10, 12] which contribute to answer this question. Let us remind that the coalescence of two graphs G 1 with distinguished vertex v 1 and G 2 with distinguished vertex v 2 , is formed by identifying vertices v 1 and v 2 that is, the vertices v 1 and v 2 are replaced by a single vertex v adjacent to the same vertices in G 1 as v 1 and the same vertices in G 2 as v 2. If it is not necessary v 1 or v 2 may not be specified. A lollipop L(p, k) is the coalescence of a cycle C p with p ≥ 3 vertices and a path P k+1 with k + 1 ≥ 2 vertices with one of its vertex of degree one as distinguished vertex, figure 1 shows an example of a lollipop. The lollipop L(p, 0) is C p. An even (resp. odd) lollipop has a cycle of even (resp. odd) length. In this paper we shall show that the lollipop graph is determined by its spectrum, answering to an open question asked in [8, 3] for even lollipop. It is known [8] that the odd lollipop is determined by its spectrum, but the 1 proof given in [8] cannot be …