In this lecture, I will present three results related to Strongly Regular Graphs. 1. An algorithm for testing isomorphism of SRGs that runs in time 2 O(√ n log n). 2. A proof that f and g, the dimensions of the eigenspaces, are both at least √ n. The problem of testing isomorphism of graphs is often reduced to the problem of giving each vertex in a graph a unique name. If we have a way of doing...

The eigenspace of the adjacency matrix of a graph possesses important information about the network structure. Recent works showed the existence of line orthogonality patterns when nodes are projected into the eigenspaces of graphs by either SVD or regular eigen decomposition. Further study proved why such phenomena hold true for simple graphs using matrix perturbation theory and several graph ...

General Berry-Esséen bounds are developed for the exponential distribution using Stein’s method and a new concentration inequality approach. As an application, a sharp error term is obtained for Hora’s result that the spectrum of the Johnson graph has an exponential limit.

Today, we will prove Cheeger’s Inequality. I consider it to be the most important theorem in spectral graph theory. Cheeger’s inequality has many variants, all of which tell us in some way that when λ2 of a graph is small, the graph has a cut of small conductance (or ratio or sparsity). Recall that in Lecture 5 we proved: Theorem 7.1.1. Let G = (V,E) be a graph and let LG be its Laplacian matri...

Disclaimer: These notes have not been subjected to the usual scrutiny reserved for formal publications. Spectral graph theory seeks to relate the eigenvectors and eigenvalues of matrices corresponding to a Graph to the combinatorial properties of the graph. While generally the theorems are designed for unweighted and undirected graphs they can be extended to the weighted graph case and (less co...

Although the hypothesis that nestedness determines mutualistic ecosystem dynamics is accepted in general, results of some recent data analyses and theoretical studies have begun to cast doubt on the impact of nestedness on ecosystem stability. However, definite conclusions have not yet been reached because previous studies are mainly based on numerical simulations. Therefore, we reveal a mathem...

We consider hexagonal systems embedded into the 3-dimensional space R3. We define the fundamental group π1(G) of such a system G and show that in case G is a finite hexagonal system with boundary, then π1(G) is a (non-abelian) free group. In this case, the rank of π1(G) equals m(G) − h(G) − n(G) + 1 where n(G) (resp. m(G), h(G)) denotes the number of vertices (resp. edges, hexagons) in G. A hex...

graph theory based methods are powerful means for representing structural systems so that their geometry and topology can be understood clearly. the combination of graph theory based methods and some metaheuristics can offer effective solutions for complex engineering optimization problems. this paper presents a charged system search (css) algorithm for the free shape optimizations of thin-wall...

The main objective of this paper is to solve the problem of finding graphs on which the spectral clustering method and the normalized cut produce different partitions. To this end, we derive formulae for minimum normalized cut for graphs in some classes such as paths, cycles, complete graphs, double-trees, lollipop graphs LPn,m, roach type graphs Rn,k and weighted paths Pn,k.

Many critical infrastructures consist of networks with complex structure, since they have highly interdependent elements and nontrivial designs with a degree of complexity that depends on the settings and level of interaction between these elements. The theory of complex networks employs techniques borrowed from graph theory to classify different types of networks, analyse their structures, and...