We show that the graph isomorphism problem is determined in a polynomial time. This is done by showing that that two graphs on n vertices are isomorphic if and only if a corresponding system of (4n− 1)n equations in n nonnegative variables is solvable. 2000 Mathematics Subject Classification: 03D15, 05C50, 05C60, 15A48, 15A51, 15A69, 90C05.

We introduce a domination polynomial of a graph G. The domination polynomial of a graph G of order n is the polynomial D(G, x) = ∑n i=γ(G) d(G, i)x , where d(G, i) is the number of dominating sets of G of size i, and γ(G) is the domination number of G. We obtain some properties of D(G, x) and its coefficients. Also we compute this polynomial for some specific graphs.

This paper introduces a trivariate graph polynomial that is a common generalization of the domination polynomial, the Ising polynomial, the matching polynomial, and the cut polynomial of a graph. This new graph polynomial, called the bipartition polynomial, permits a variety of interesting Research supported by ESF. Research supported by the Austrian National Research Network S11403-N23 (RiSE) ...

In this paper, we introduce a novel graph polynomial called the 'information polynomial' of a graph. This graph polynomial can be derived by using a probability distribution of the vertex set. By using the zeros of the obtained polynomial, we additionally define some novel spectral descriptors. Compared with those based on computing the ordinary characteristic polynomial of a graph, we perform ...

The matching polynomial of a graph is the generating function of the numbers of its matchings with respect to their cardinality. A graph polynomial is polynomial reconstructible, if its value for a graph can be determined from its values for the vertex-deleted subgraphs of the same graph. This note discusses the polynomial reconstructibility of the matching polynomial. We collect previous resul...

The Martin polynomial of an oriented Eulerian graph encodes information about families of cycles in the graph. This paper uses a transformation of the Martin polynomial that facilitates standard combinatorial manipulations. These manipulations result in several new identities for the Martin polynomial, including a di.erentiation formula. These identities are then applied to get new combinatoria...

the eigenvalues of a graph is the root of its characteristic polynomial. a fullerene f is a 3-connected graphs with entirely 12 pentagonal faces and n/2 -10 hexagonal faces, where n isthe number of vertices of f. in this paper we investigate the eigenvalues of a class of fullerenegraphs.

The reconstruction conjecture has remained open for simple undirected graphs since it was suggested in 1941 by Kelly and Ulam. In an attempt to prove the conjecture, many graph invariants have been shown to be reconstructible from the vertex-deleted deck, and in particular, some prominent graph polynomials. Among these are the Tutte polynomial, the chromatic polynomial and the characteristic po...