The degree sequences and spectra of scale‐free random graphs

We investigate the degree sequences of scale‐free random graphs. We obtain a formula for the limiting proportion of vertices with degree d, confirming non‐rigorous arguments of Dorogovtsev, Mendes, and Samukhin ( 14 ). We also consider a generalization of the model with more randomization, proving similar results. Finally, we use our results on the degree sequence to show that for certain values of parameters localized eigenfunctions of the adjacency matrix can be found. © 2005 Wiley Periodicals, Inc. Random Struct. Alg., 2006     

Some results on the convex hull of finitely many convex sets

A better than quadratic estimate is given for the volume of the convex hull of points on Hadamard manifolds with pinched curvature. It was known previously that the volume is bounded by some polynomial in . The estimate comes from the study of the convex hull of finitely many convex sets on Hadamard manifolds.

     

The polytope of degree sequences of hypergraphs

Let D_n(r) denote the convex hull of degree sequences of simple r-uniform hypergraphs on the vertex set {1,2,…,n}. The polytope D_n(2) is a well-studied object. Its extreme points are the threshold sequences (i.e., degree sequences of threshold graphs) and its facets are given by the Erdös–Gallai inequalities. In this paper we study the polytopes D_n(r) and obtain some partial information. Our approach also yields new, simple proofs of some basic results on D_n(2). Our main results concern the extreme points and facets of D_n(r). We characterize adjacency of extreme points of D_n(r) and, in the case r=2, determine the distance between two given vertices in the graph of D_n(2). We give a characterization of when a linear inequality determines a facet of D_n(r) and use it to bound the sizes of the coefficients appearing in the facet defining inequalities; give a new short proof for the facets of D_n(2); find an explicit family of Erdös–Gallai type facets of D_n(r); and describe a simple lifting procedure that produces a facet of D_n+1(r) from one of D_n(r).     

Counting degree sequences of spanning trees in bipartite graphs: A graph-theoretic proof

Given a bipartite graph with bipartition each spanning tree in has a degree sequence on and one on . Löhne and Rudloff showed that the number of possible degree sequences on equals the number of possible degree sequences on . Their proof uses a non-trivial characterization of degree sequences by -draconian sequences based on polyhedral results of Postnikov. In this paper, we give a purely graph-theoretic proof of their result.     

The signed domatic number of some regular graphs

Let G be a finite and simple graph with vertex set V(G), and let f:V(G)→{−1,1} be a two-valued function. If ∑_x∈N[v]f(x)≥1 for each v∈V(G), where N[v] is the closed neighborhood of v, then f is a signed dominating function on G. A set {f₁,f₂,…,f_d} of signed dominating functions on G with the property that for each x∈V(G), is called a signed dominating family (of functions) on G. The maximum number of functions in a signed dominating family on G is the signed domatic number on G. In this paper, we investigate the signed domatic number of some circulant graphs and of the torus C_p×C_q.     

A characterization of projective-planar signed graphs

A signed graph has a plus or minus sign on each edge. A simple cycle is positive or negative depending on whether it contains an even or odd number of negative edges, respectively. We consider embeddings of a signed graph in the projective plane for which a simple cycle is essential if and only if it is negative. We characterize those signed graphs that have such a projective-planar embedding. Our characterization is in terms of a related signed graph formed by considering the theta subgraphs in the given graph.     

Colouring of generalized signed triangle-free planar graphs

We view an undirected graph $G$ as a symmetric digraph, where each edge $xy$ is replaced by two opposite arcs $e=(x,y)$ and $e^{?1}=(y,x)$. Assume $S$ is an inverse closed subset of permutations of positive integers. We say $G$ is $S$-$k$-colourable if for any mapping $\sigma :E\left(G\right)\to S$ with $\sigma (x,y)={\left(\sigma (y,x)\right)}^{?1}$, there is a mapping $f:V\left(G\right)\to \left[k\right]=\{1,2,\dots ,k\}$ such that ${\sigma }_e\left(f\left(x\right)\right)\ne f\left(y\right)$ for each arc $e=(x,y)$. The concept of $S$-$k$-colourable is a common generalization of several other colouring concepts. This paper is focused on finding the sets $S$ such that every triangle-free planar graph is $S$-3-colourable. Such a set $S$ is called TFP-good. Grötzsch’s theorem is equivalent to say that $S=\left\{id\right\}$ is TFP-good. We prove that for any inverse closed subset $S$ of $S_3$ which is not isomorphic to $\{id,\left(12\right)\}$, $S$ is TFP-good if and only if either $S=\left\{id\right\}$ or there exists $a\in \left[3\right]$ such that for each $\pi \in S$, $\pi \left(a\right)\ne a$. It remains an open question to determine whether or not $S=\{id,\left(12\right)\}$ is TFP-good.     

The signed star domination numbers of the Cartesian product graphs

Let G be a graph with vertex set V(G) and edge set E(G). A function f:E(G)→{-1,1} is said to be a signed star dominating function of G if for every v∈V(G), where E_G(v)={uv∈E(G)|u∈V(G)}. The minimum of the values of , taken over all signed star dominating functions f on G, is called the signed star domination number of G and is denoted by γ_SS(G). In this paper, a sharp upper bound of γ_SS(G×H) is presented.     

The number of graphs and a random graph with a given degree sequence

We consider the set of all graphs on n labeled vertices with prescribed degrees D = (d₁,…,d_n). For a wide class of tame degree sequences D we obtain a computationally efficient asymptotic formula approximating the number of graphs within a relative error which approaches 0 as n grows. As a corollary, we prove that the structure of a random graph with a given tame degree sequence D is well described by a certain maximum entropy matrix computed from D. We also establish an asymptotic formula for the number of bipartite graphs with prescribed degrees of vertices, or, equivalently, for the number of 0‐1 matrices with prescribed row and column sums. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2013     

The convex hull of two core capacitated network design problems

The network loading problem (NLP) is a specialized capacitated network design problem in which prescribed point-to-point demand between various pairs of nodes of a network must be met by installing (loading) a capacitated facility. We can load any number of units of the facility on each of the arcs at a specified arc dependent cost. The problem is to determine the number of facilities to be loaded on the arcs that will satisfy the given demand at minimum cost.This paper studies two core subproblems of the NLP. The first problem, motivated by a Lagrangian relaxation approach for solving the problem, considers a multiple commodity, single arc capacitated network design problem. The second problem is a three node network; this specialized network arises in larger networks if we aggregate nodes. In both cases, we develop families of facets and completely characterize the convex hull of feasible solutions to the integer programming formulation of the problems. These results in turn strengthen the formulation of the NLP.Research of this author was supported in part by a Faculty Grant from the Katz Graduate School of Business, University of Pittsburgh.     

Some performance tests of convex hull algorithms

The two-dimensional convex hull algorithms of Graham, Jarvis, Eddy, and Akl and Toussaint are tested on four different planar point distributions. Some modifications are discussed for both the Graham and Jarvis algorithms. Timings taken of FORTRAN implementations indicate that the Eddy and Akl-Toussaint algorithms are superior on uniform distributions of points in the plane. The Graham algorithm outperforms the others on those distributions where most of the points are on or near the boundary of the hull.     

Decreasing the diameter of bounded degree graphs

Let f_d (G) denote the minimum number of edges that have to be added to a graph G to transform it into a graph of diameter at most d. We prove that for any graph G with maximum degree D and n > n₀ (D) vertices, f₂(G) = n − D − 1 and f₃(G) ≥ n − O(D³). For d ≥ 4, f_d (G) depends strongly on the actual structure of G, not only on the maximum degree of G. We prove that the maximum of f_d (G) over all connected graphs on n vertices is n/⌊d/2 ⌋ − O(1). As a byproduct, we show that for the n‐cycle C_n, f_d (C_n) = n/(2⌊d/2 ⌋ − 1) − O(1) for every d and n, improving earlier estimates of Chung and Garey in certain ranges. © 2000 John Wiley & Sons, Inc. J Graph Theory 35: 161–172, 2000     

A simple criterion on degree sequences of graphs

A finite sequence of nonnegative integers is called graphic if the terms in the sequence can be realized as the degrees of vertices of a finite simple graph. We present two new characterizations of graphic sequences. The first of these is similar to a result of Havel-Hakimi, and the second equivalent to a result of Erd?s & Gallai, thus providing a short proof of the latter result. We also show how some known results concerning degree sets and degree sequences follow from our results.     

A short proof of a theorem on degree sets of graphs

For a finite simple graph G, we denote the set of degrees of its vertices, known as its degree set, by D(G). Kapoor, Polimeni and Wall [Degree sets for graphs, Fund. Math. 95 (1977) 189-194] have determined the least number of vertices among graphs with a given degree set. We give a very short proof of this result.     

The expected convex hull trimmed regions of a sample

Given a data set in the multivariate Euclidean space, we study regions of central points built by averaging all their subsets with a fixed number of elements. The averaging of these sets is performed by appropriately scaling the Minkowski or elementwise summation of their convex hulls. The volume of such central regions is proposed as a multivariate scatter estimate and a circular sequence algorithm to compute the central regions of a bivariate data set is described. Extended version of the conference paper Cascos (2006) presented at the 17th Conference of the European Regional Section of the International Association for Statistical Computing (COMPSTAT 2006, Rome, Italy, August 28–September 1, 2006). Supported by the Spanish Ministry of Education and Science under grant MTM2005-02254.     

The convex hull of a Banach–Saks set

A subset A of a Banach space is called Banach–Saks when every sequence in A has a Cesàro convergent subsequence. Our interest here focuses on the following problem: is the convex hull of a Banach–Saks set again Banach–Saks? By means of a combinatorial argument, we show that in general the answer is negative. However, sufficient conditions are given in order to obtain a positive result.     

Asymptotics of the convex hull of spherically symmetric samples

In this paper we consider the convex hull of a spherically symmetric sample in R^d. Our main contributions are some new asymptotic results for the expectation of the number of vertices, number of facets, area and the volume of the convex hull assuming that the marginal distributions are in the Gumbel max-domain of attraction. Further, we briefly discuss two other models assuming that the marginal distributions are regularly varying or O-regularly varying.