رضا شرف دینی

Let $G$ be a simple undirected graph with the vertex set $ V (G)=\left\{ {v_1,\ldots , v_n} \right\}$. Denote by $d_i$ the degree of the vertex $v_i$. The adjacency matrix $A(G)$ of $G$ is a square matrix of order $n$, whose $(i,j)$-entry is 1 if $v_i$ and $v_j$ are adjacent in $G$ and $0$ otherwise. The degree matrix $D(G)$ of $G$ is a diagonal matrix of order $n$ defined as $D(G)=\mathrm{diag}(d_1,\ldots,d_n)$. The matrices $ L(G)=D(G)-A(G) $ and $ Q(G)=D(G) A(G) $ are called the Laplacian matrix and the signless Laplacian matrix of $G$, respectively. The multiset of eigenvalues of $A(G)$ (resp. $\mathcal{L}(G)$, $Q(G)$) is called the $A$-spectrum (resp. $\mathcal{L}$-spectrum, $Q$-spectrum) of $G$. The join of two graphs $G$ and $H$, is a graph formed from disjoint copies of $G$ and $H$ by connecting each vertex of $G$ to each vertex of $H$. The join of a regular graph with a complete graph is called a multicone graph. In this thesis we determine some classes of multicone graphs by their $A$-spectrum, $\mathcal{L}$-spectrum and $Q$-spectrum.