Algebraic Connectivity for Vertex-Deleted Subgraphs, and a Notion of Vertex Centrality

Let G be a connected graph, suppose that v is a vertex of G, and denote the subgraph formed from G by deleting vertex v by G\v. Denote the algebraic connectivities of G and G\v by α(G) and (G\v), respectively. In this paper, we consider the functions ∅(v) = α(G)− α(G\v) and k(v) = α(G\v)/α(G), provide attainable upper and lower bounds on both functions, and characterise the equality cases in those bounds. The function yields a measure of vertex centrality, and we apply that measure to analyse certain graphs arising from food webs

Node and Edge Differential Privacy for Graph Laplacian Spectra: Mechanisms and Scaling Laws

This paper develops a framework for privatizing the spectrum of the graph Laplacian of an undirected graph using differential privacy. We consider two privacy formulations. The first obfuscates the presence of edges in the graph and the second obfuscates the presence of nodes. We compare these two privacy formulations and show that the privacy formulation that considers edges is better suited to most engineering applications. We use the bounded Laplace mechanism to provide differential privacy to the eigenvalues of a graph Laplacian, and we pay special attention to the algebraic connectivity, which is the Laplacian's second smallest eigenvalue. Analytical bounds are presented on the the accuracy of the mechanisms and on certain graph properties computed with private spectra. A suite of numerical examples confirms the accuracy of private spectra in practice.Comment: arXiv admin note: text overlap with arXiv:2104.0065

Introdução à teoria espectral de grafos

O objetivo principal desta dissertação é fazer uma introdução à teoria espectral de grafos. Abordaremos algumas propriedades dos grafos, nomeadamente o polinómio característico, os valores e vetores próprios de matrizes associadas aos grafos. Nesta dissertação iremos dar mais relevância à matriz de adjacência e à matriz laplaciana, fazendo uma análise de alguns tipos especí…cos de grafos e dos respetivos espectros. Iremos estudar o conceito de energia de um grafo e de medidas de centralidade associadas aos grafos e aos seus conceitos inerentes. Serão apresentadas duas aplicações no decorrer da dissertação, uma referente à Química e ao carbono quaternário, com o objetivo de descobrir se o carbono quaternário existe ou não na molécula em estudo e, outra referente às medidas de centralidade em que será feito a análise do jogo de futebol “Portugal vs França” a contar para a …nal do Europeu de 2016, com o intuito de descobrir as performances dos jogadores.The main purpose of this dissertation is to make an introduction to the spectral graph theory. We will study some properties of graphs, namely the characteristic polynomial, the eigenvalues and the eigenvectors speci…c to matrices associated with graphs. In this dissertation, we will give more importance to the adjacency matrix and to the Laplacian matrix and we will do an analysis of some speci…c types of graphs and their respective spectra. We will study the energy of a graph and the measures of centrality associated to graphs and their inherent concepts. Two practical applications will be presented throughout the dissertation, one related to chemistry and quaternary carbon, in order to …nd out whether or not the quaternary carbon exists in the molecule under study, and another one related to the centrality measures in which the analysis of the football match “Portugal versus France”, that de…ned the European Champion of 2016 will be done, with the objective of sorting out the performances of the players on the …eld