Extremal Values of the Interval Number of a Graph

The interval number $i( G )$ of a simple graph $G$ is the smallest number $t$ such that to each vertex in $G$ there can be assigned a collection of at most $t$ finite closed intervals on the real line so that there is an edge between vertices $v$ and $w$ in $G$ if and only if some interval for $v$ intersects some interval for $w$. The well known interval graphs are precisely those graphs $G$ with $i ( G )\leqq 1$. We prove here that for any graph $G$ with maximum degree $d, i ( G )\leqq \lceil \frac{1}{2} ( d + 1 ) \rceil$. This bound is attained by every regular graph of degree $d$ with no triangles, so is best possible. The degree bound is applied to show that $i ( G )\leqq \lceil \frac{1}{3}n \rceil$ for graphs on $n$ vertices and $i ( G )\leqq \lfloor \sqrt{e} \rfloor$ for graphs with $e$ edges

Locally ss-distance transitive graphs

We give a unified approach to analysing, for each positive integer $s$, a class of finite connected graphs that contains all the distance transitive graphs as well as the locally $s$-arc transitive graphs of diameter at least $s$. A graph is in the class if it is connected and if, for each vertex $v$, the subgroup of automorphisms fixing $v$ acts transitively on the set of vertices at distance $i$ from $v$, for each $i$ from 1 to $s$. We prove that this class is closed under forming normal quotients. Several graphs in the class are designated as degenerate, and a nondegenerate graph in the class is called basic if all its nontrivial normal quotients are degenerate. We prove that, for $s\geq 2$, a nondegenerate, nonbasic graph in the class is either a complete multipartite graph, or a normal cover of a basic graph. We prove further that, apart from the complete bipartite graphs, each basic graph admits a faithful quasiprimitive action on each of its (1 or 2) vertex orbits, or a biquasiprimitive action. These results invite detailed additional analysis of the basic graphs using the theory of quasiprimitive permutation groups.Comment: Revised after referee report

A Statistical Mechanical Load Balancer for the Web

The maximum entropy principle from statistical mechanics states that a closed system attains an equilibrium distribution that maximizes its entropy. We first show that for graphs with fixed number of edges one can define a stochastic edge dynamic that can serve as an effective thermalization scheme, and hence, the underlying graphs are expected to attain their maximum-entropy states, which turn out to be Erdos-Renyi (ER) random graphs. We next show that (i) a rate-equation based analysis of node degree distribution does indeed confirm the maximum-entropy principle, and (ii) the edge dynamic can be effectively implemented using short random walks on the underlying graphs, leading to a local algorithm for the generation of ER random graphs. The resulting statistical mechanical system can be adapted to provide a distributed and local (i.e., without any centralized monitoring) mechanism for load balancing, which can have a significant impact in increasing the efficiency and utilization of both the Internet (e.g., efficient web mirroring), and large-scale computing infrastructure (e.g., cluster and grid computing).Comment: 11 Pages, 5 Postscript figures; added references, expanded on protocol discussio

Operads, configuration spaces and quantization

We review several well-known operads of compactified configuration spaces and construct several new such operads, C, in the category of smooth manifolds with corners whose complexes of fundamental chains give us (i) the 2-coloured operad of A-infinity algebras and their homotopy morphisms, (ii) the 2-coloured operad of L-infinity algebras and their homotopy morphisms, and (iii) the 4-coloured operad of open-closed homotopy algebras and their homotopy morphisms. Two gadgets - a (coloured) operad of Feynman graphs and a de Rham field theory on C - are introduced and used to construct quantized representations of the (fundamental) chain operad of C which are given by Feynman type sums over graphs and depend on choices of propagators.Comment: 58 page