82,438 research outputs found
Balanced and 1-balanced graph constructions
AbstractThere are several density functions for graphs which have found use in various applications. In this paper, we examine two of them, the first being given by b(G)=|E(G)|/|V(G)|, and the other being given by g(G)=|E(G)|/(|V(G)|−ω(G)), where ω(G) denotes the number of components of G. Graphs for which b(H)≤b(G) for all subgraphs H of G are called balanced graphs, and graphs for which g(H)≤g(G) for all subgraphs H of G are called 1-balanced graphs (also sometimes called strongly balanced or uniformly dense in the literature). Although the functions b and g are very similar, they distinguish classes of graphs sufficiently differently that b(G) is useful in studying random graphs, g(G) has been useful in designing networks with reduced vulnerability to attack and in studying the World Wide Web, and a similar function is useful in the study of rigidity. First we give a new characterization of balanced graphs. Then we introduce a graph construction which generalizes the Cartesian product of graphs to produce what we call a generalized Cartesian product. We show that generalized Cartesian product derived from a tree and 1-balanced graphs are 1-balanced, and we use this to prove that the generalized Cartesian products derived from 1-balanced graphs are 1-balanced
An asymptotic bound for the strong chromatic number
The strong chromatic number of a graph on
vertices is the least number with the following property: after adding isolated vertices to and taking the union with any
collection of spanning disjoint copies of in the same vertex set, the
resulting graph has a proper vertex-colouring with colours.
We show that for every and every graph on vertices with
, , which is
asymptotically best possible.Comment: Minor correction, accepted for publication in Combin. Probab. Compu
The Evolution of Beliefs over Signed Social Networks
We study the evolution of opinions (or beliefs) over a social network modeled
as a signed graph. The sign attached to an edge in this graph characterizes
whether the corresponding individuals or end nodes are friends (positive links)
or enemies (negative links). Pairs of nodes are randomly selected to interact
over time, and when two nodes interact, each of them updates its opinion based
on the opinion of the other node and the sign of the corresponding link. This
model generalizes DeGroot model to account for negative links: when two enemies
interact, their opinions go in opposite directions. We provide conditions for
convergence and divergence in expectation, in mean-square, and in almost sure
sense, and exhibit phase transition phenomena for these notions of convergence
depending on the parameters of the opinion update model and on the structure of
the underlying graph. We establish a {\it no-survivor} theorem, stating that
the difference in opinions of any two nodes diverges whenever opinions in the
network diverge as a whole. We also prove a {\it live-or-die} lemma, indicating
that almost surely, the opinions either converge to an agreement or diverge.
Finally, we extend our analysis to cases where opinions have hard lower and
upper limits. In these cases, we study when and how opinions may become
asymptotically clustered to the belief boundaries, and highlight the crucial
influence of (strong or weak) structural balance of the underlying network on
this clustering phenomenon
Syntactic Separation of Subset Satisfiability Problems
Variants of the Exponential Time Hypothesis (ETH) have been used to derive lower bounds on the time complexity for certain problems, so that the hardness results match long-standing algorithmic results. In this paper, we consider a syntactically defined class of problems, and give conditions for when problems in this class require strongly exponential time to approximate to within a factor of (1-epsilon) for some constant epsilon > 0, assuming the Gap Exponential Time Hypothesis (Gap-ETH), versus when they admit a PTAS. Our class includes a rich set of problems from additive combinatorics, computational geometry, and graph theory. Our hardness results also match the best known algorithmic results for these problems
- …