Jason I. Brown

Roots of Independence Polynomials of Well Covered Graphs

Let G be a well covered graph, that is, all maximal independent sets of G have the same cardinality, and let ik denote the number of independent sets of cardinality k in G. We investigate the roots of the independence polynomial i(G, x) = ∑ ikxk. In particular, we show that if G is a well covered graph with independence number β, then all the roots of i(G, x) lie in in the disk |z| ≤ β (this is far from true if the condition of being well covered is omitted). Moreover, there is a family of well covered graphs (for each β) for which the independence polynomials have a root arbitrarily close to −β.

Roots of the reliability polynomial

The reliability of a graph G is the probability that G is connected, given that edges are independently operational with probability p. This is known to be a polynomial in p, and the location of the roots of these functions is discussed. In particular, it is conjectured that the roots of the reliability polynomial of any connected graph lie in the disc

On the Location of Roots of Independence Polynomials

| z - 1 | \leq 1

On Generalized Graph Colorings

, and evidence for this conjecture is provided. It is shown that all real roots lie in

On the Chromatic Roots of Generalized Theta Graphs

\{ 0 \} \cup ( 1,2 ]

On chromatic roots of large subdivisions of graphs

and that every graph has a subdivision for which the roots of the reliability polynomial lie in the conjectured disc.

Non-Stanley Bounds for Network Reliability

The independence polynomial of a graph G is the function i(G, x) = ∑k≥0ikxk, where ik is the number of independent sets of vertices in G of cardinality k. We prove that real roots of independence polynomials are dense in (−∞, 0], while complex roots are dense in ℂ, even when restricting to well covered or comparability graphs. Throughout, we exploit the fact that independence polynomials are essentially closed under graph composition.

Network transformations and bounding network reliability

Given a property P, graph G, and k ⩾ 0, a P k-coloring is a function π: V(G) {1, …, k} such that the subgraph induced by each color class has property P; χ(G : P) is the least k, for which G has a P k-coloring. We investigate here the theory of P colorings. Generalizations of the wellknown notions of vertex criticality and unique colorability are discussed, and we extend a theorem of Erdos to P chromatic graphs.

Chromatic polynomials and order ideals of monomials

The generalized theta graph ?s1, ?, sk consists of a pair of endvertices joined by k internally disjoint paths of lengths s1, ?, sk?1. We prove that the roots of the chromatic polynomial ?(?s1, ?, sk, z) of a k-ary generalized theta graph all lie in the disc|z?1|?1+o(1)]k/logk, uniformly in the path lengths si. Moreover, we prove that ?2, ?, 2?K2, k indeed has a chromatic root of modulus 1+o(1)]k/logk. Finally, for k?8 we prove that the generalized theta graph with a chromatic root that maximizes |z?1| is the one with all path lengths equal to 2; we conjecture that this holds for all k.

The complexity of generalized graph colorings

Abstract Given a graph G, we derive an expression for the chromatic polynomials of the graphs resulting from subdividing some (or all) of its edges. For special subfamilies of these, we are able to describe the limits of their chromatic roots. We also prove that for any e >0, all sufficiently large subdivisions of G have their chromatic roots in | z −1| e . A consequence of our work will be a characterization of the graphs having a subdivision whose chromatic polynomial has a root with negative real part.