We describe the asymptotic properties of the edge-triangle exponential random
graph model as the natural parameters diverge along straight lines. We show
that as we continuously vary the slopes of these lines, a typical graph drawn
from this model exhibits quantized behavior, jumping from one complete
multipartite graph to another, and the jumps happen precisely at the normal
lines of a polyhedral set with infinitely many facets. As a result, we provide
a complete description of all asymptotic extremal behaviors of the model.Comment: 38 pages, 7 figure
The birthday paradox states that there is at least a 50% chance that some two out of twenty-three randomly chosen people will share the same birth date. The calculation for this problem assumes that all birth dates are equally likely. We consider the following two modifications of this question. If the distribution of birthdays is non-uniform, does that increase or decrease the probability of matching birth dates? Further, what if we focus on birthdays shared by some particular pairs rather than any two people. Does a non-uniform distribution on birth dates increase or decrease the probability of a matching pair? In this paper we present our results in this generalized setting. We use some results and methods due to Sokal [17] concerning bounds on the roots of chromatic polynomials to prove our results.
Abstract. Let P G (q) denote the chromatic polynomial of a graph G on n vertices. The 'shameful conjecture' due to Bartels and Welsh states that,Let µ(G) denote the expected number of colors used in a uniformly random proper n-coloring of G. The above inequality can be interpreted as saying that µ(G) ≥ µ(On), where On is the empty graph on n nodes. This conjecture was proved by F. M. Dong, who in fact showed that,There are examples showing that this inequality is not true for all q ≥ 2. In this paper, we show that the above inequality holds for all q ≥ 36D 3/2 , where D is the largest degree of G. It is also shown that the above inequality holds true for all q ≥ 2 when G is a claw-free graph.
The chromatic polynomial is a well studied object in graph theory. There are many results and conjectures about the log-concavity of the chromatic polynomial and other polynomials related to it. The location of the roots of these polynomials has also been well studied. One famous result due to A. Sokal and C. Borgs provides a bound on the absolute value of the roots of the chromatic polynomial in terms of the highest degree of the graph. We use this result to prove a modification of a log-concavity conjecture due to F. Brenti. The original conjecture of Brenti was that the chromatic polynomial is log-concave on the natural numbers. This was disproved by Paul Seymour by presenting a counter example. We show that the chromatic polynomial P G (q) of graph G is in fact log-concave for all q > C∆ + 1 for an explicit constant C < 10, where ∆ denotes the highest degree of G. We also provide an example which shows that the result is not true for constants C smaller than 1.
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