‎‎let r be a non-commutative ring with unity‎. ‎the commuting graph‎ ‎of $r$ denoted by $gamma(r)$‎, ‎is a graph with a vertex set‎ ‎$rsetminus z(r)$ and two vertices $a$ and $b$ are adjacent if and only if‎ ‎$ab=ba$‎. ‎in this paper‎, ‎we investigate non-commutative rings with unity of order $p^n$ where $p$ is prime and $n in lbrace 4,5 rbrace$‎. it is shown that‎, ‎$gamma(r)$ is the disjoint ...

In evolutionary graph theory biologists study the problem of determining the probability that a small number of mutants overtake a population that is structured on a weighted, possibly directed graph. Currently Monte Carlo simulations are used for estimating such fixation probabilities on directed graphs, since no good analytical methods exist. In this paper, we introduce a novel deterministic ...

We explore under what conditions simple combinatorial attributes and algorithms such as the distance sequence and degree-based partitioning and refinement can be used to distinguish vertices of inhomogeneous random graphs. In the classical setting of Erdős-Renyi graphs and random regular graphs it has been proven that vertices can be distinguished in a constant number of rounds of degree-based ...

Active contour models based on partial differential equations have proved successful in image segmentation, yet the study of their geometric formulation on arbitrary geometric graphs is still at an early stage. In this paper, we introduce geometric approximations of gradient and curvature, which are used in the geodesic active contour model. We prove convergence in probability of our gradient a...

Given a class of graphs G closed under taking minors, we study the maximum degree ∆n of random graphs from G with n vertices. We prove several lower and upper bounds that hold with high probability. Among other results, we find classes of graphs providing orders of magnitude for ∆n not observed before, such us log n/ log log log n and log n/ log log log log n. © 2016 Elsevier Ltd. All rights re...

We study the biased random walk where at each step of a "controller" can, with certain small probability, move to an arbitrary neighbour. This model was introduced by Azar et al. [STOC'1992]; we extend their work time dependent setting and consider cover times this walk. obtain new bounds on hitting times. conjectured that controller can increase stationary probability vertex from $p$ $p^{1-\ep...

Bisimplicial edges in bipartite graphs are closely related to pivots in Gaussian elimination that avoid turning zeroes into non-zeroes. We present a new deterministic algorithm to find such edges in bipartite graphs. The expected time complexity of our new algorithm is O ( n2 log n ) on random bipartite graphs in which each edge is present with a fixed probability p, a polynomial improvement ov...

Inspired by previous work of Diaz, Petit, Serna, and Trevisan (Approximating layout problems on random graphs Discrete Mathematics, 235, 2001, 245–253), we show that several well-known graph layout problems are approximable to within a factor arbitrarily close to 1 of the optimal with high probability for random graphs drawn from an ErdösRenyi distribution with appropriate sparsity conditions. ...

The probability distribution on a set S = { 1, 2, . . . , n } defined by Pr(k) = 1/(Hnk), where Hn in the nth harmonic number, is commonly called a Zipfian distribution. In this note we look at the degree distribution of graphs created from graphs with Zipfian degree distributions using some standard graph theoretical operations. Introduction: The probability distribution on a set S = { 1, 2, ....

We investigate flows on graphs whose links have random capacities. For binary trees we derive the probability distribution for the maximal flow from the root to a leaf, and show that for infinite trees it vanishes beyond a certain threshold that depends on the distribution of capacities. We then examine the maximal total flux from the root to the leaves. Our methods generalize to simple graphs ...