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The degree-diameter problem seeks to find the largest possible number of vertices in a graph having given diameter and given maximum degree. Very often the problem is studied for restricted families of graph such as vertex-transitive or…
The inverse degree of a graph is the sum of the reciprocals of the degrees of its vertices. We prove that in any connected planar graph, the diameter is at most 5/2 times the inverse degree, and that this ratio is tight. To develop a…
In new progress on conjectures of Stein, and Addario-Berry, Havet, Linhares Sales, Reed and Thomass\'e, we prove that every oriented graph with all in- and out-degrees greater than 5k/8 contains an alternating path of length k. This…
Let $F$ be a probability distribution with support on the non-negative integers. Four methods for generating a simple undirected graph with (approximate) degree distribution $F$ are described and compared. Two methods are based on the so…
This article gives an overview of the emerging literature on large deviations for random graphs. Written for the general mathematical audience, the article begins with a short introduction to the theory of large deviations. This is followed…
Exploring small connected and induced subgraph patterns (CIS patterns, or graphlets) has recently attracted considerable attention. Despite recent efforts on computing the number of instances a specific graphlet appears in a large graph…
In this paper, we present a hybrid graph-drawing algorithm (GDA) for layouting large, naturally-clustered, disconnected graphs. We called it a hybrid algorithm because it is an implementation of a series of already known graph-drawing and…
In 2000, Cohn, Kenyon and Propp studied uniformly random perfect matchings of large induced subgraphs of $\mathbb Z^2$ (a.k.a. dimer configurations or domino tilings) and developed a large deviation theory for the associated height…
Here I report about the modifications of and relations between graphic lambda calculus, various formalisms which appeared under the name chemlambda and a version of directed interaction combinators. This is part of the study and experiments…
Continuous spline functions are defined as piecewise polynomials on the faces of a polyhedral complex that agree on the intersections of two faces. Splines are used in approximation theory and numerical analysis, with applications in data…
We analyze the distribution of the distance between two nodes, sampled uniformly at random, in digraphs generated via the directed configuration model, in the supercritical regime. Under the assumption that the covariance between the…
We present techniques for maintaining subgraph frequencies in a dynamic graph, using data structures that are parameterized in terms of h, the h-index of the graph. Our methods extend previous results of Eppstein and Spiro for maintaining…
A survey article for AMS Summer Institute at Seattle in 2005.
Erd\H{o}s proved that there are graphs with arbitrarily large girth and chromatic number. We study the extension of this for generalized chromatic numbers.
We improve recent results relating graph eigenvalues to other graph parameters like girth, domination number, and minimum degree.
In 1974, Erd\H{o}s asked the following question: given a graph $G$ and a directed graph $\vec{H}$, how many ways are there to orient the edges of $G$ such that it does not contain $\vec{H}$ as a subgraph? We denote this value by $D(G,…
Motivated by very large-scale communication networks, we newly introduce exponentiation of graphs. Using the exponential operation on graphs, we can construct various graphs of multi-exponential order with logarithmic diameter. We show that…
Refereed version to appear on Journal of Algebraic Geometry. Unchanged content. Strongly ameliorated presentation thanks to referee's action.
We generate new mathematical tools with which to quantify the macroscopic topological structure of large directed networks. This is achieved via a statistical mechanical analysis of constrained maximum entropy ensembles of directed random…
In this paper, we determine the maximum size of a nonhamiltonian-connected graph with prescribed order and minimum degree. We also characterize the extremal graphs that attain this maximum size. This work generalizes a previous result…