Reza Sharafdini

A simple connected graph G with n vertices and m edges is said to be characterized by its spectrum if there exist no other non-isomorphic graph that is cospectral to G. In this thesis, it was shown that for any arbitrary strongly regular multicone graph, can be characterized by its adjacency spectrum, with an example Peterson graph as an exam- ple. We investigated also, the spectral characterization of the union of a Tree, several copies of the complete graph on one and two vertices, and the bipartite graph, T ∪ rK1 ∪ sK2 ∪ t1K1,p1−1 ∪ tsK1,ps−1 by their both Laplacian and signless Laplacian matrix. Some spectral properties were presented and some graphs which are Determined by their signless Laplacian Spectrum (DQS) were constructed from some known DQS graphs in the literature. In this thesis also, we constructed some graph- energy-invariants based on the Weiner and Hosoya matrix of a molecular graph, WH(G) and established some relationships between them and the adjacency matrix. Some bounds of the energy were investigated. A code for the computation of Wiener-Hosoya matrix of graphs was formulated in SAGEMath software. The topological indices of some chemical com- pounds of anticancer and anticoronavirus drugs was investigated, and the Quantitative Structural Property Relationship analysis was carried out to predict some physical properties of those drugs. The vertex-weighted signless Laplacian energy of a simple connected graph G was also studied and some relationship with the Laplacian energy with vertex-weight was also considered