These notes are intended to supplement the lecture slides and existing notes and references on graphs available at our institution (in particular Jeff Erickson’s excellent notes) and else where. I assume that the reader is already familiar with the basics of graphs and their representation via adjacency lists and adjaceny matrices. We will asssume throughout these notes that graphs are represen...

In this paper, we use analysis on graphs to study quantitative measures of segregation. We focus a classical statistic from the geography and urban sociology literature known as Moran’s , which in our language is score associated real-valued function graph, computed with respect spatial weight matrix such adjacency geographic units that tile city. Our results characterizing extremal behavior il...

altan derivatives of polycyclic conjugated hydrocarbons were recently introduced and studied in theoretical organic chemistry. we now provide a generalization of the altan concept, applicable to any graph. several earlier noticed topological properties of altan derivatives of polycyclic conjugated hydrocarbons are shown to be the properties of all altan derivatives of all graphs. among these ar...

We study the eigenvectors and eigenvalues of random matrices with iid entries. Let N be a matrix entries which have symmetric distribution. For each unit eigenvector v our main results provide small ball probability bound for linear combinations coordinates v. Our generalize works Meehan Nguyen [59] as well Touri second author [67, 68, 69] matrices. Along way, we an optimal estimate that has si...

‎Let $Gamma$ be a graph with adjacency eigenvalues $lambda_1leqlambda_2leqldotsleqlambda_n$‎. ‎Then the energy of‎ ‎$Gamma$‎, ‎a concept defined in 1978 by Gutman‎, ‎is defined as $mathcal{E}(G)=sum_{i=1}^n|lambda_i|$‎. ‎Also‎ ‎the Estrada index of $Gamma$‎, ‎which is defined in 2000 by Ernesto Estrada‎, ‎is defined as $EE(Gamma)=sum_{i=1}^ne^{lambda_i}$‎. ‎In this paper‎, ‎we compute the eigen...

The characteristic polynomial of the adjacency matrix of a graph is noted in connection with a quantity characterizing the topological nature of structural isomers saturated hydrocarbons [S], a set of numbers that are the same for all graphs isomorphic to the graph, and others [l]. Many properties of the characteristic polynomials of the adjacency matrices of a graph and its line graph [3] have...

Given a graph G; let kGk denote the trace norm of its adjacency matrix, also known as the energy of G: The main result of this paper states that if G is a graph of order n; then kGk + G (n 1) 1 +n ; where G is the complement of G: Equality is possible if and only if G is a strongly regular graph with parameters (n; (n 1) =2; (n 5) =4; (n 1) =4) ; known also as a conference graph. In fact, the a...

Randić index is one of the most famous topological graph indices. The energy a was defined more than four decades ago for its molecular applications. classical modeling molecule as sum absolute values all eigenvalues adjacency matrix corresponding to graph. There are several other versions notion obtained using types matrices. In this paper, we introducing and investigating type Hadi RHE(G) G, ...

Symmetry in mathematical programming may lead to a multiplicity of solutions. In nonconvex optimisation, it can negatively affect the performance of the Branch and Bound algorithm. Symmetry may induce large search trees with multiple equivalent solutions, i.e. with the same optimal value. Dealing with symmetry requires detecting and classifying it first. This paper develops several methods for ...

Regarding the adjacency matrices of n-vertex graphs and related graph Laplacian, we introduce two families of discrete matrix models constructed both with the help of the Erdős-Rényi ensemble of random graphs. Corresponding matrix sums represent the characteristic functions of the average number of q-step walks and q-step closed walks over the random graph. These sums can be considered as the d...