Some Results On Graphs Determined By Their Spectra

The question "Which graphs are determined by their spectra?" goes back for half a century, and originates from chemistry. In 1956, Günthard and Primas raised the question in a paper that relates the theory of graph spectra to Hückel's theory from chemistry. At that time, all graphs were considered to be determined by their spectra. But, a pair of cospectral graphs were found out by Collatz and Sinogowitz a year later. In 1966, Fisher who considered a question of Kac:"Can one hear the shape of a drum?" modelled the shape of the drum by a graph. Then, the sound of that drum is characterized by the eigenvalues of the graph. Thus, Kac's question is essentially ours'. A graph is said to be determined by its spectrum if there is no other non-isomorphic graph with the same spectrum. If two or more graphs have the same spectrum, then these graphs are cospectral graphs. Thus, finding cospectral graphs belongs to the question "Which graphs are determined by their spectra?" Many cospectral graphs were found out after Collatz and Sinogowitz found out the pair of cospectral graphs. But, answering this problem seems out of research, now.In this thesis, some graphs with special structure are considered. The lollipop graph, denoted by H_n,p, is obtained by appending a cycle C_p to a pendant vertex ofa path P_n-p. A multi-fan graph is a graph of the form (P_n1 + P_n2+…+P_nk×b,where b is a universal vertex, and P_n1+ P_n2+…+P_nk is the disjoint union ofpaths P_ni(n₁≥1) for i = 1,2,…, k. Note that the particular case of k = 1 in the definition is just the classical fan graph F_(n1)+1. Similar to the multi-fan graph, the multii-wheel graph is the graph (C_n1 + C_n2 +…+ C_nk)×b, where C_n1 + C_n2+…+ C_nk is the disjoint union of circuits C_ni, and k≥1 and n_i≥3 for i = 1,2,…, k. Note that the particular case of k = 1 in the definition is just the classical single wheel graph W_(n1)+1. A tree is called double starlike if it has exactly two vertices of degree greater than two. We denote by H_n(p,p) (n≥2,p≥1) one special double starlike graph. In this thesis, we prove that(1) lollipop graph H_n,p with p odd is determined by its adjacency spectrum.(2) graph H′_n,p with p odd is determined by its adjacency spectrum.(3) lollipop graph H_n,p is determined by its Laplacian spectrum.(4) lollipop graph H_n,p is determined by its Q-spectrum.(5) muti-fan graph is determined by its Laplacian spectrum.(6) odd single wheel graph is determined by its Laplacian spectrum. (7) odd multi-wheel graph is determined by its Laplacian spectrum.(8) graph H_n(p,p) is determined by its Laplacian spectrum.