Polynomial Invariants of Graphs

We define two polynomials f(G) and /*(G) for a graph G by a recursive formula with respect to deformation of graphs. Analyzing their various properties, we shall discuss when two graphs have the same polynomials. Introduction. Our graphs are finite, undirected ones with or without self-loops and multiple edges. As elementary deformations of graphs, we have the deletion and the contraction of edges. Especially we define the contraction of an edge e as an operation which deletes e and identifies its ends to a single vertex. If an edge e is a loop, its deletion and contraction are identical. We denote the resulting graph of deletion and contraction of an edge e G E(G) by G — e and G/e, respectively. We define a polynomial f(G;t,x,y) for a graph G as follows. (We shall often simply write f(G).) (i) f(K~n) =tn(n> 1). (ii) f(G)_ = xf(G/e) + yf(G e) (e G E(G)). The graph Kn is the complement of the complete graph Kn over n vertices, that is, it consists of only n isolated vertices. The repetition of deleting and contracting edges transforms finally into Kn, so f(G) can be determined by the above two formulas. The well-definedness of f(G) will be shown in Theorem 1.1. It is convenient to set /(0) = 1 for the null graph 0 which has no vertex and no edge. If there is an isomorphism a: G —> H between two graphs G and H, then f(G) and f(H) can be calculated in parallel via o and f(G) = f(H). So this polynomial f(G) is an isomorphism invariant. As is shown in §2, f(G) includes various information about a graph G, namely the numbers of subgraphs, cutsets, vertex-colorings, flows and so on. So f(G) seems to be a very useful invariant for the isomorphism problem of graphs. However there are many graphs whose isomorphism types cannot be distinguished by f(G). For example, all trees with the same number of edges, say q, have the same polynomial t(x + ty)". Our arguments in the last half of this paper will show that f(G) is a 2-isomorphism invariant rather than an isomorphism invariant. The definition of 2-isomorphism can be found in [5] and will be given in §5. Roughly speaking, two graphs are said to be 2-isomorphic if they can be transformed into each other by certain kinds of modifications at 1or 2-vertex cuts. (Our 2-isomorphism is slightly different from the ordinary one. We do not regard two graphs to be 2-isomorphic when one is obtained from the other by breaking it at a cut vertex. Under our definition, 2-isomorphic graphs have the same number of components.) In particular, when a Received by the editors February 12, 1986. 1980 Mathematics Subject Classification (1985 Revision). Primary 05C99.