let $r$ be a commutative ring with identity. let $g(r)$ denote the maximal graph associated to $r$, i.e., $g(r)$ is a graph with vertices as the elements of $r$, where two distinct vertices $a$ and $b$ are adjacent if and only if there is a maximal ideal of $r$ containing both. let $gamma(r)$ denote the restriction of $g(r)$ to non-unit elements of $r$. in this paper we study the various graphi...

A class of graphs is bridge-addable if given a graph G in the class, any graph obtained by adding an edge between two connected components of G is also in the class. The authors recently proved a conjecture of McDiarmid, Steger, and Welsh stating that if G is bridge-addable and Gn is a uniform n-vertex graph from G, then Gn is connected with probability at least (1 + o(1))e−1/2. The constant e−...

This paper presents a convergence analysis for the problem of consistent labelling using genetic search. The work builds on a recent empirical study of graph matching where we showed that a Bayesian consistency measure could be e$ciently optimised using a hybrid genetic search procedure which incorporated a hill-climbing step. In the present study we return to the algorithm and provide some the...

Given a sample from a probability measure with support on a submanifold in Euclidean space one can construct a neighborhood graph which can be seen as an approximation of the submanifold. The graph Laplacian of such a graph is used in several machine learning methods like semi-supervised learning, dimensionality reduction and clustering. In this paper we determine the pointwise limit of three d...

Existing approaches to analyzing the asymptotics of graph Laplacians typically assume a well-behaved kernel function with smoothness assumptions. We remove the smoothness assumption and generalize the analysis of graph Laplacians to include previously unstudied graphs including kNN graphs. We also introduce a kernel-free framework to analyze graph constructions with shrinking neighborhoods in g...

We consider the problem of approximating the PageRank of a target node using only local information provided by a link server. This problem was originally studied by Chen, Gan, and Suel (CIKM 2004), who presented an algorithm for tackling it. We prove that local approximation of PageRank, even to within modest approximation factors, is infeasible in the worst-case, as it requires probing the li...

We consider a family of compact manifolds which shrinks with respect to an appropriate parameter to a graph. The main result is that the spectrum of the Laplace-Beltrami operator converges to the spectrum of the (differential) Laplacian on the graph with Kirchhoff boundary conditions at the vertices. On the other hand, if the shrinking at the vertex parts of the manifold is sufficiently slower ...

Graph Shift (GS) algorithms are recently focused as a promising approach for discovering dense subgraphs in noisy data. However, there are no theoretical foundations for proving the convergence of the GS Algorithm. In this paper, we propose a generic theoretical framework consisting of three key GS components: simplex of generated sequence set, monotonic and continuous objective function and cl...

The Robust Graph mode seeking by Graph Shift [1] (RGGS) algorithm represents a recent promising approach for discovering dense subgraphs in noisy data. However, there are no theoretical foundations for proving the convergence of the RGGS algorithm, leaving the question as to whether an algorithm works for solid reasons. In this paper, we propose a generic theoretical framework consisting of thr...