5 research outputs found
Note on the smallest root of the independence polynomial
One can define the independence polynomial of a graph G as follows. Let i(k)(G) denote the number of independent sets of size k of G, where i(0)(G) = 1. Then the independence polynomial of G is I(G,x) = Sigma(n)(k=0)(-1)(k)i(k)(G)x(k). In this paper we give a new proof of the fact that the root of I(G,x) having the smallest modulus is unique and is real
On a conjecture of Wilf
Let n and k be natural numbers and let S(n,k) denote the Stirling numbers of
the second kind. It is a conjecture of Wilf that the alternating sum
\sum_{j=0}^{n} (-1)^{j} S(n,j) is nonzero for all n>2. We prove this conjecture
for all n not congruent to 2 and not congruent to 2944838 modulo 3145728 and
discuss applications of this result to graph theory, multiplicative partition
functions, and the irrationality of p-adic series.Comment: 18 pages, final version, accepted for publication in the Journal of
Combinatorial Theory, Series
Generalizations of the Matching Polynomial to the Multivariate Independence Polynomial
We generalize two main theorems of matching polynomials of undirected simple
graphs, namely, real-rootedness and the Heilmann-Lieb root bound. Viewing the
matching polynomial of a graph as the independence polynomial of the line
graph of , we determine conditions for the extension of these theorems to
the independence polynomial of any graph. In particular, we show that a
stability-like property of the multivariate independence polynomial
characterizes claw-freeness. Finally, we give and extend multivariate versions
of Godsil's theorems on the divisibility of matching polynomials of trees
related to .Comment: This research was supported by NSF Grant CCF-155375, 20 page