Reza Sharafdini

Let {$G = (V, E)$} be a simple graph of order {$n$} and size {$m$}, with vertex set {$V (G) = \{v1, v2,\dots,vn\}$}, without isolated vertices and sequence of vertex degrees, {$\Delta=d_{1}\geq d_{2}\geq \dots \geq d_{n}=\delta>0$}, {$d_i=d_G(v_i)$}. If the vertices {$v_i$} and {$v_j$} are adjacent, we denote it as {$v_i v_j \in E(G)$} or {$i \sim j$}. With {$TI$} we denote a topological index that can be represented as {$TI=TI(G)=\sum_{i\sim j}\mathcal{F}(d_{i},d_{j})$}, where $\mathcal{F}$ is an appropriately chosen function with the property {$\mathcal{F}(x,y)=\mathcal{F}(y,x)$}. A general extended adjacency matrix {$A=(a_{ij})$} of $G$ is defined as {$a_{ij}=\mathcal{F}(d_{i}, d_{j})$} if the vertices $v_i$ and $v_j$ are adjacent, and {$a_{ij}=0$} otherwise. Denote by {$f_i$}, {$ i = 1, 2,\dots,n$} the eigenvalues of {$A$}. The energy of the general extended adjacency matrix is defined as {$\mathcal{E}_{TI}=\mathcal{E}_{TI}(G)=\sum_{i=1}^n |f_{i}|$} . In this thesis, first, we present various new upper bounds for the adjacency energy of graphs in terms of several graph invariants such as the number of vertices, number of edges, maximum degree and Zagreb indices of the graph. We also characterize graphs achieving equality in each new bound. Second, lower and upper bounds on {$\mathcal{E}_{TI}$} are obtained. By means of the present approach a plethora of earlier established results can be obtained as special cases.