Reza Sharafdini

Let $G$ be a connected graph with the vertex set $V(G)=\{v_1,\ldots,v_n\}$. The distance $d_G(u,v)$ between vertices $u$ and $v$ of a connected graph $G$ is equal to the number of edges in a minimal path connecting them. The transmission of the vertex $v_i\in V(G)$, denoted by $\sigma_G(v_i)$, is defined to be the sum of distances between $v_i$ and any other vertices in $G$ , i.e., $\sigma_G(v_i)=\sum_{j=1}^n{{{d}_{G}}(v_i,v_j)}$. A topological index is said to be a transmission-based topological index (TT index) if it includes the transmissions $\sigma(u)$ of vertices of $G$. In this thesis our aim was i) to define various types of transmission-based topological indices ii) establish lower and upper bounds for them, iii) determine a family of graphs for which these bounds are best possible, iv) we show in examples that using a group theoretical approach the transmission-based topological indices can be easily computed for a particular set of regular, vertex-transitive, and edge-transitive graphs, and v) we have an spectral approach to some of this indices. The Laplacian transmission matrix of $G$ is defined as $L_{Tr}(G)=\diag(\sigma_G(v_1),\cdots,\sigma_G(v_n))-A(G)$, where $A(G)$ is the adjacency matrix of $G$. We show that $L_{Tr}(G)$ is positive semidefinite. Let $\mu_1^{\prime},\cdots,\mu_n^{\prime}$ be the eigenvalues of $L_{Tr}(G)$. Then a transmission version of Laplacian energy of $G$ is defined as $EL_{Tr}(G)=\sum_{i=1}^n\Big|\mu_i^{\prime} - \frac{2W(G)}{n}\Big|$. We obtain some bounds for $EL_{Tr}(G)$ in terms of other invariants of $G$ like, ordinary energy, Wiener index and variable transmission Zagreb index.