Reza Sharafdini

Let $G$ be a graph with a corresponding matrix $M(G)$. The multiset of eigenvalues of $M(G)$ is called $M$-spectrum of $G$. Two graphs are said to be $M$-cospectral if they have the same $M$-spectrum. A graph is said to be determined by its $M$-spectrum if there is no other non-isomorphic graphs $M$-cospectral with it. Let $A(G)$ and $D(G)$ be the adjacency and the degree matrix of $G$. The Laplacian and signless Laplacian matrix of $G$ are defined as $\mathcal{L}(G)=D(G)-A(G)$ and $Q(G) = A(G)+ D(G)$, respectively. A tree is called double starlike if it has exactly two vertices of degree greater than two. Specially, a double starlike tree obtained by attaching $p\ge 2$ pendant vertices to an end vertex of the path $P_n, (n\ge 2)$ and attaching $q\ge 2$ pendant vertices to another end of the path $P_n$ is denoted by $H_n(p,q)$. In this thesis, we first aim to obtain some bounds for Laplacian and signless Laplacian eigenvalues of graphs. Then we prove that $H_n(p,q)$ is determined by its Laplacian and signless Laplacian spectrum.