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We prove tight upper bounds on the logarithmic derivative of the independence and matching polynomials of d-regular graphs. For independent sets, this theorem is a strengthening of the results of Kahn, Galvin and Tetali, and Zhao showing that a union of copies of $K_{d,d}$ maximizes the number of independent sets and the independence polynomial of a d-regular graph. For matchings, this shows that the matching polynomial and the total number of matchings of a d-regular graph are maximized by a union of copies of $K_{d,d}$. Using this we prove the asymptotic upper matching conjecture of Friedland, Krop, Lundow, and Markstrom. In probabilistic language, our main theorems state that for all d-regular graphs and all $lambda$, the occupancy fraction of the hard-core model and the edge occupancy fraction of the monomer-dimer model with fugacity $lambda$ are maximized by $K_{d,d}$. Our method involves constrained optimization problems over distributions of random variables and applies to all d-regular graphs directly, without a reduction to the bipartite case.
We determine the maximum number of maximal independent sets of arbitrary graphs in terms of their covering numbers and we completely characterize the extremal graphs. As an application, we give a similar result for Konig-Egervary graphs in terms of their matching numbers.
Grinblat (2002) asks the following question in the context of algebras of sets: What is the smallest number $mathfrak v = mathfrak v(n)$ such that, if $A_1, ldots, A_n$ are $n$ equivalence relations on a common finite ground set $X$, such that for ea
There are distributed graph algorithms for finding maximal matchings and maximal independent sets in $O(Delta + log^* n)$ communication rounds; here $n$ is the number of nodes and $Delta$ is the maximum degree. The lower bound by Linial (1987, 1992)
In this paper, we study independent domination in directed graphs, which was recently introduced by Cary, Cary, and Prabhu. We provide a short, algorithmic proof that all directed acyclic graphs contain an independent dominating set. Using linear alg
The notion of a Riordan graph was introduced recently, and it is a far-reaching generalization of the well-known Pascal graphs and Toeplitz graphs. However, apart from a certain subclass of Toeplitz graphs, nothing was known on independent sets in Ri