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