Graph-functions associated with an edge-property

Let P be an edge-property of graphs. For any graph G we construct a polynomial Ψ(G, η,P), in an indeterminate η, in which the coefficient of η for any r ≥ 0 gives the number of subsets of E(G) that have cardinality r and satisfy P . An example is the well known matching polynomial of a graph. After studying the properties of Ψ(G, η,P) in general, we specialise to two particular edge-properties: that of being an edge-covering and that of inducing an acyclic subgraph. The resulting polynomials, called the edge-cover and acyclic polynomials respectively, are studied and recursive formulae for computing them are derived. As examples we calculate these polynomials for paths and cycles. 1 A graph-function related to an edge-property The graphs considered in this paper are undirected finite non-null graphs which may contain multiple edges and loops. For a graph G, let V (G), E(G), v(G) and e(G) be the vertex set, edge set, order and size of G respectively. An edge-property P of graphs is a property possessed by some edge sets, provided the following condition is satisfied: ∗ This paper was mostly completed when Dong was a post-doctoral fellow at Massey University from 1998–2000. He thanks Massey University for its financial support. 4 DONG, HENDY, TEO AND LITTLE for any graphs G1 and G2 with G1 ∼= G2, if E1 ⊆ E(G1) corresponds to E2 ⊆ E(G2) under an isomorphism, then E1 has property P in G1 if and only if E2 has property P in G2. Examples of edge-properties are property Pc that the subgraph induced by the edges is spanning, property Pa that the subgraph induced by the edges is acyclic, and property Pm that the subgraph induced by the edges is a matching. The analogous concept of vertex-properties has been studied in [1]. Let P be any edge-property and G be any graph. Define P(G) = {E ′ ⊆ E(G)|E ′ has edge-property P in G}. For any integer n ≥ 0, define F(G, n,P) to be the set of mappings f : {1, 2, · · · , n} → E(G), subject to the condition that {f(1), f(2), · · · , f(n)} ∈ P(G). Note that when n = 0, {f(1), f(2), · · · , f(n)} is empty. We write F (G, n,P) = |F(G, n,P)|. (1) Observe that F (G, n,P) is a graph-function. Lemma 1.1 For any edge-property P and graph G, F (G, 0,P) = { 1, if ∅ ∈ P(G), 0, otherwise. (2) An edge-property P is said to be inclusive if P(H) ⊆ P(G) for any graph G and spanning subgraph H of G. For a ∈ E(G), let G−a denote the graph obtained from G by deleting a. We write F(G, a, n,P) = F(G, n,P)− F(G− a, n,P), (3) and F (G, a, n,P) = |F(G, a, n,P)|. (4) Note that F(G, a, n,P) is the set of f in F(G, n,P) such that f−1(a) = ∅. Lemma 1.2 Let P be an inclusive edge-property. Then F (G, a, n,P) = F (G, n,P)− F (G− a, n,P). (5) Proof. Since P is inclusive, we have F(G− a, n,P) ⊆ F(G, n,P). Hence the result holds. For two graphs G1, G2, let G1 ⊕ G2 be the graph H with a vertex partition {V1, V2} such that H[Vi] ∼= Gi for i = 1, 2, and x and y are not adjacent for any GRAPH-FUNCTIONS ASSOCIATED WITH AN EDGE-PROPERTY 5 x ∈ V1 and y ∈ V2. For a disconnected graph G with two subgraphs G1 and G2 such that V (G1) ∩ V (G2) = ∅ and E(G) = E(G1) ∪ E(G2), we have G ∼= G1 ⊕G2. An edge-property P is said to be resolvable if for any graph G = G1 ⊕ G2 and E ′ ⊆ E(G), E ′ ∈ P(G) if and only if E ′ ∩E(G1) ∈ P(G1) and E ′ ∩E(G2) ∈ P(G2). Theorem 1.1 Let P be a resolvable edge-property. For graphs G1 and G2, F (G1 ⊕G2, n,P) = n ∑ r=0 ( n r ) F (G1, r,P)F (G2, n− r,P). (6) Proof. Let G = G1 ⊕G2. For any mapping f from {1, 2, · · · , n} into E(G), let N1 = {1 ≤ k ≤ n|f(k) ∈ E(G1)} and N2 = {1, 2, · · · , n} −N1. Define two mappings g1 : N1 → E(G1) and g2 : N2 → E(G2), gi(k) = f(k), k ∈ Ni, i = 1, 2. Since P is resolvable, {f(1), f(2), · · · , f(n)} ∈ P(G) if and only if g1(N1) ∈ P(G1) and g2(N2) ∈ P(G2). On the other hand, for any partition {N1, N2} of {1, 2, · · · , n} and any mappings g1, g2: g1 : N1 → E(G1) and g2 : N2 → E(G2), we can define a mapping f : {1, 2, · · · , n} → E(G) given by f(k) = gi(k) if k ∈ Ni for i = 1, 2. Given a partition {N1, N2} of {1, 2, · · · , n} with |N1| = r, there are F (G1, r,P) mappings g1 from N1 into E(G1) and F (G2, n − r,P) mappings g2 from N2 into E(G2) such that g1(N1) ∈ P(G1) and g2(N2) ∈ P(G2). Thus there are F (G1, r,P)F (G2, n− r,P) different mappings f from {1, 2, · · · , n} into E(G) such that {i|f(i) ∈ E(G1)} = N1 and {f(1), f(2), · · · , f(n)} ∈ P(G). Hence F (G1 ⊕G2, n,P) = ∑ N1∪N2={1,2,···,n} N1∩N2=∅ F (G1, |N1|,P)F (G2, |N2|,P)