Consider the following stochastic process on a graph: initially all vertices are uncovered and at each step cover the two vertices of a random edge. What is the expected number of steps required to cover all vertices of the graph? In this note we show that the mean cover time for a regular graph of N vertices is asymptotically (N log N)/2. Moreover, we compute the exact mean cover time for some...

The decycling number (G) of a graph G is the smallest number of vertices which can be removed from G so that the resultant graph contains no cycles. In this paper, we study the decycling numbers of random regular graphs. For a random cubic graph G of order n, we prove that (G) n/4 1/ 2 holds asymptotically almost surely. This is the result of executing a greedy algorithm for decycling G makin...

Under the assumption of proportional hazards, the log-rank test is optimal for testing the null hypothesis of H0 : β = 0, where β denotes the logarithm of the hazard ratio. However, if there are additional covariates that correlate with survival times, making use of their information will increase the efficiency of the log-rank test. We apply the theory of semiparametrics to characterize a clas...

It was only recently shown by Shi and Wormald, using the differential equation method to analyse an appropriate algorithm, that a random 5-regular graph asymptotically almost surely has chromatic number at most 4. Here, we show that the chromatic number of a random 5-regular graph is asymptotically almost surely equal to 3, provided a certain four-variable function has a unique maximum at a giv...

Some variants of the (block) Gauss–Seidel iteration for solution linear systems with M-matrices in Hessenberg form are discussed. Comparison results asymptotic convergence rate some regular splittings derived: particular, we prove that a lower-Hessenberg M-matrix $$\rho (P_{GS})\ge \rho (P_S)\ge (P_{AGS})$$ , where $$P_{GS}, P_S, P_{AGS}$$ matrices Gauss–Seidel, staircase, and anti-Gauss–Seidel...

We show that on every Ramanujan graph G, the simple random walk exhibits cutoff: when G has n vertices and degree d, the total-variation distance of the walk from the uniform distribution at time t = d d−2 logd−1 n + s √ logn is asymptotically P(Z > c s) where Z is a standard normal variable and c = c(d) is an explicit constant. Furthermore, for all 1 ≤ p ≤ ∞, d-regular Ramanujan graphs minimiz...

The purpose of this paper is to provide the new demicloseness principle– τ (weakly) demicloseness principle. We prove that if X is a Banach space with locally uniformly τ -Opial condition, where τ is a Hausdorff topology on X, C is a nonempty τ compact subset of X, and T : C → C is a asymptotically nonexpansive type mapping. If {xα} is a net in C which converges to x in the sense of τ topology ...

Rho kinase mediates the effects of inflammatory permeability factors by increasing actomyosin-generated traction forces on endothelial adherens junctions, resulting in disassembly of intercellular junctions and increased vascular leakage. In vitro, this is accompanied by the Rho kinase-driven formation of prominent radial F-actin fibers, but the in vivo relevance of those F-actin fibers has bee...

An upper bound is given on the minimum distance between i subsets of the same size of a regular graph in terms of the i-th largest eigenvalue in absolute value. This yields a bound on the diameter in terms of the i-th largest eigenvalue, for any integer i. Our bounds are shown to be asymptotically tight. A recent result by Quenell relating the diameter, the second eigenvalue, and the girth of a...