Generalizing a previous one-variable " interlace polynomial " , we consider a new interlace polynomial in two variables. The polynomial can be computed in two very different ways. The first is an expansion analogous to the state space expansion of the Tutte polynomial; the differences are that our expansion is over vertex rather than edge subsets, the rank of the subset appears positively rathe...

Many graph polynomials, such as the Tutte polynomial, the interlace polynomial and the matching polynomial, have both a recursive definition and a defining subset expansion formula. In this article, we present a general, logic-based framework which gives a precise meaning to recursive definitions of graph polynomials. We then prove in this framework that every recursive definition of a graph po...

A connected partition of a graph G is a partition of its vertex set such that each induced subgraph is connected. In earlier research it was called a connected composition of a graph. We find a polynomial, the defect polynomial of the graph, that decribes the number of connected partitions of complements of graphs with respect to any complete graph. The defect polynomial is calculated for sever...

One of the more exciting polynomials among newly presented graph algebraic is M−Polynomial, which a standard method for calculating degree−based topological indices. In this paper, we define Mve−polynomials based on vertex edge degree and derive various vertex–edge indices from them. Thus, any graph, provide some relationships between Also, discuss general Mve−polynomial r−regular simple graph....

A graph polynomial P (G, x) is called reconstructible if it is uniquely determined by the polynomials of the vertex deleted subgraphs of G for every graph G with at least three vertices. In this note it is shown that subgraph-counting graph polynomials of increasing families of graphs are reconstructible if and only if each graph from the corresponding defining family is reconstructible from it...

It is well-known that the Jones polynomial of a knot is closely related to the Tutte polynomial of a special graph obtained from a regular projection of the knot. In this paper, we study the Tutte polynomials for signed graphs. We show that if a signed graph is constructed from a simpler graph via k-thickening or k-stretching, then its Tutte polynomial can be expressed in terms of the Tutte pol...

It is known that a certain case of the all-terminal network reliability can be computed via the Tutte polynomial which is an invariant in graph theory. Recently, we have proposed a new approach of computing the Tutte polynomial of a graph of moderate size [9]. In this paper, by using it we analyze computation method of the general all-terminal network reliability and two-terminal network reliab...

The energy of signed graph is the sum of the absolute values of the eigenvalues of its adjacency matrix. Two signed graphs are said to be equienergetic if they have same energy. In the literature the construction of equienergetic signed graphs are reported. In this paper we obtain the characteristic polynomial and energy of the join of two signed graphs and thereby we give another construction ...

We prove that if a graph H has the same Tutte polynomial as the line graph of a d-regular, d-edge-connected graph, then H is the line graph of a d-regular graph. Using this result we prove that the line graph of a regular complete t-partite graph is uniquely determined by its Tutte polynomial. We prove the same result for the line graph of any complete bipartite graph.

A geometric graph is a labeled graph whose vertices are points in the 2D plane with an isomorphism invariant under geometric transformations such as translation, rotation, and scaling. While Kuramochi and Karypis (ICDM2002) extensively studied the frequent pattern mining problem for geometric subgraphs, the maximal graph mining has not been considered so far. In this paper, we study the maximal...