The neighbourhood polynomial of a graph

We examine neighbourhood polynomials, which are generating functions for the number of faces of each cardinality in the neighbourhood complex of a graph. We provide explicit polynomials for hypercubes, for graphs not containing a four-cycle and for the graphs resulting from joins and Cartesian products. We also show that the closure of the roots are dense in the complex plane except possibly in the disc |z + 1| < 1. 1 The Neighbourhood Polynomial of a Graph There are a number of graph polynomials that have been widely studied. Chromatic polynomials count the number of proper colourings of a graph. Matching polynomials enumerate matchings. Independence polynomials are generating polynomials for the number of independent sets of each cardinality. (All Terminal) reliability polynomials provide the probability of communication between all pairs of vertices given that edges are independently operation with the same probability. For each polynomial there is an underlying complex, which puts all of these polynomials in a common framework. A (simplicial) complex on a finite set X is a collection C of subsets of X, closed under containment. Each set in C is called a face of the complex, and the maximal faces (with respect to containment) are called facets or bases. The dimension of a complex C is the maximum cardinality of a face. The f–vector (or face–vector) of a d–dimensional complex C is (f0, f1, . . . , fd), where fi is the number of faces of cardinality i in C. Finally the f–polynomial of a d–dimensional complex C is the generating function fC(x) = ∑ i fix i for the f–vector (f0, f1, . . . , fd) of the complex. For each of the previously described graph polynomials, there is a complex for which the graph polynomial is a simple evaluation of the f–polynomial. The independence complex I(G) of graph G is the complex on the vertex set V of G whose faces are the independent sets of G. The independence polynomial is merely the f– polynomial of the independence complex. The complex on the edge set of G whose ∗ Both authors wish to thank the NSERC for partial financial support. 56 JASON I. BROWN AND RICHARD J. NOWAKOWSKI faces are those sets of edges whose removal leaves G connected is called the cographic matroid of G, Cog(G), and the reliability polynomial of G is in fact Rel(G, p) = pmfCog(G) ( 1− p p ) , where m is the number of edges of G. Given a linear order < of the edges E of G, the broken circuit complex, BC(G,<) is the complex on E whose faces are subsets of the edges that don’t contain a broken circuit (that is, a circuit minus its <-least edge). Then π(G, x) = xfBC(G,<)(−1/x), where n is the number of vertices of G. One of the most startling applications of simplicial complexes to graph theory is undoubtedly Lovász’s proof [25] of the chromatic number of Kneser graphs. His argument centers on the neighbourhood complex N (G) of a graph, whose vertices are the vertices of the graph and whose faces are subsets of vertices that have a common neighbour. We define a univariate polynomial, which we call the neighbourhood polynomial of graph G: neighG(x) = ∑ U∈N (G) x|U |. While there have been a number of articles that explored properties of neighbourhood complexes [2, 16, 24, 25], the neighbourhood polynomial has not been previously investigated. As an example, in a four-cycle {a, b, c, d}, the empty set trivially has a common neighbour (as the graph has at least one vertex) while each of the single vertices has a neighbour. Each set {a, c} and {b, d} has two common neighbours, but one suffices, and there is no subset of three vertices that have a common neighbour. Thus the neighbourhood complex is {∅, {a}, {b}, {c}, {d}, {a, b}, {b, d}} and the associated neighbourhood polynomial is neighC4(x) = 1 + 4x+ 2x . Note that this is the same as the neighbourhood polynomial for a path with four vertices, so, as might be expected, these polynomials are not unique to the underlying graph. As another example, note that neighKn(x) = (1 + x) n − x, as every subset of the vertices of a complete graph except the entire vertex set has a common neighbour. Similarly, the neighbourhood polynomial of the complete bipartite graph Km,n is (1+x) m+(1+x)n−1 (as the first two terms have overcounted the empty set). The facets (i.e., maximal faces) of the neighbourhood complex of a graph G are simply the neighbourhoods of each vertex. Thus the number of facets is unusually THE NEIGHBOURHOOD POLYNOMIAL OF A GRAPH 57 small for a complex—it is at most the number of vertices in the graph and moreover, they can be enumerated in polynomial time. This contrasts sharply with all of the other graph polynomials mentioned earlier. The difficulty in determining the neighbourhood complex and polynomials lies in the recognition of the overcounting of subsets. Let N1, . . . , Nk be the maximal (with respect to containment) neighbourhoods of the vertices of a graph G with n vertices and m edges. In general, k ≤ n as some vertices may have the same neighbourhoods, or one might be a subset of the other. A set belongs to the neighbourhood complex ofG if and only if it is a subset of one of the Ni’s. Thus inclusion-exclusion can be used to enumerate such sets, and to generate the neighbourhood polynomial. Unfortunately, the algorithm is exponential, so it is of little use in calculations. On closer examination, and assuming that G has no isolated vertices, a first order approximation for the neighbourhood polynomial is ∑ v∈V (1 + x)deg(v) − x ∑ v∈V (deg(v)− 1)− (n− 1) = ∑ v∈V (1 + x)deg(v) − x(2m − n)− (n− 1) where deg(v) is the degree of v. This corresponds to counting all subsets of each of the facets (i.e. neighbourhoods) and correcting for counting the empty set n times and each vertex v deg(v) many times. Those sets that are in the neighbourhoods of two or more vertices are also overcounted but it is more difficult to see how to easily correct the formula. But for one class of graphs, the formula is correct as it stands. We say G is C4-free if G does not contain C4 as a subgraph (not necessarily induced). If G is C4-free then two vertices can have at most one common neighbour and therefore the only non-empty sets that have more than one common neighbour are the singletons. This argument proves the following result. Theorem 1 Let G be C4-free with n vertices and m edges. Then neighG(x) = ∑ v∈V (1 + x)deg(v) − x(2m − n)− (n− 1). An immediate observation is the following. Corollary 2 The neighbourhood polynomial for a C4-free graph depends only on the degree sequence of the graph and can be calculated in polynomial time. Theorem 1 gives us the neighbourhood polynomials for many graphs, including: • If G = Cn, a cycle of length n > 4, then neighG(x) = 1 + nx+ nx; • If G is an r-regular graph of girth at least 5, neighG(x) = n(1+x)r−n(r−1)x− (n−1). In particular, for the Petersen graph, neighG(x) = 1+10x+30x+10x; • If G is a tree, then neighG(x) = ∑ v∈V (1 + x) deg(v) − x(n− 1)− (n− 1). 58 JASON I. BROWN AND RICHARD J. NOWAKOWSKI 2 Graph Operations and Neighbourhood Polynomials What is the effect that various graph operations have on the neighbourhood polynomial? The first is trivial, relying solely on the fact that the only sets of the disjoint union of two graphs having a common neighbour are the sets in each of the graphs that have a common neighbour, with the empty set being the only such set the graphs have in common. Lemma 3 Let G and H be graphs and G ∪ H be the disjoint union of G and H. Then neighG∪H(x) = neighG(x) + neighH(x) − 1.