The Study Of Spectral Determination Of Two Families Of Cyclic Graph

The spectral determination theory of graph is a new field in graph theory, which ismainly concerned with the adjacency spectrum, the Laplacian spectrum and the signlessLaplacian spectrum.The question"which graphs are determined by their spectra?"goes back for half acentury, and originates from chemistry. In1956, Gu¨nthard and Primas raised the questionin a paper that relates the theory of graph spectra to Hu¨ckel’s theory from chemistry. Atthat time it was believed that every graph is determined by its spectrum until one yearlater Collatz and Sinogowitz presented a pair of cospectral graphs. Now, many cospectralgraphs have been found out.A graph is said to be determined by its spectrum if there is no other non-isomorphicgraph with the same spectrum. If two or more graphs have the same spectrum, then thesegraphs are cospectral graphs. Thus, finding cospectral graphs belongs to the question"Which graphs are determined by their spectra?" Many cospectral graphs were foundout after Collatz and Sinogowitz found out the pair of cospectral graphs. But, answeringthis problem seems out of research, now.Spectral graph theory has some important applications in fields such as physics,chemistry, computer science and so on, and it is in progress.In this paper, we study the Laplacian spectra of two class of non-regular graphs, getthe following results:1. Study the Laplacian spectrum of graph H(n; q, n1, n2, n3), prove that graph H(n; q,n1, n2, n3) is determined by its Laplacian spectrum.2. Study the Laplacian spectrum of dumbbell graph d(p, q, r), prove that if two dumb-bell graphs have the same Laplacian spectrum then they must be isomorphic.3. Apply the relation between the Laplacian spectrum of bipartite graph and the adja-cency spectrum of its line graph, prove that graph H(n; q, n1, n2, n3) with q even isdetermined by its Laplacian spectrum.