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....

Various modularity matrices appeared in the recent literature on network analysis and algebraic graph theory. Their purpose is to allow writing as quadratic forms certain combinatorial functions appearing in the framework of graph clustering problems. In this paper we put in evidence certain common traits of various modularity matrices and shed light on their spectral properties that are at the...

Determining the Fiedler vector of the Laplacian or adjacency matrices of graphs is the most computationally intensive component of several applications, such as graph partitioning, graph coloring, envelope reduction, and seriation. Often an approximation of the Fiedler vector is suÆcient. We discuss issues involved in the use of Monte Carlo techniques for this purpose.

Graovac and Pisanski [On the Wiener index of a graph, J. Math. Chem. 8 (1991) 53 – 62] applied an algebraic approach to generalize the Wiener index by symmetry group of the molecular graph under consideration. In this paper, exact formulas for this graph invariant under some graph operations are presented.

The number of spanning trees of a graph G is called the complexity of G and denoted c(G). A classical result in algebraic graph theory explicitly diagonalizes the Laplacian of the n-cube C(n) and yields, using the Matrix-Tree theorem, an explicit formula for c(C(n)). In this paper we explicitly block diagonalize the Laplacian of the q-analog Cq(n) of C(n) and use this, along with the Matrix-Tre...

The algebraic analysis of Petri Nets combines structural analysis (Transition and Place-Invariants) with information about the initial marking of the net. The main advantage of algebraic analysis over graph based methods is, that the problem of state space explosion is avoided. Hence, algebraic conditions are-in most cases-easier to check for large systems. In this contribution necessary and su...

This contribution follows the line of research to formally deene a visual language by algebraic concepts and techniques. In this paper, a graphic i.e. a sentence of a visual language, is equipped with an underlying graph structure. This is done in a way that each graph object of the graph structure is represented by a graphic which may be a complex one. Graph structures are regarded as n-sorted...

We obtain Poisson equations satisfied by elliptic modular graph functions with four links. Analysis of these leads to a non--trivial algebraic relation between the various graphs.

In his study of Newton's root approximation method, Smale (1985) deened the Newto-nian graph of a complex univariate polynomial f. The vertices of this graph are the roots of f and f 0 and the edges are the degenerate curves of ow of the Newtonian vector eld N f (z) = ?f(z)=f 0 (z). The embedded edges of this graph form the boundaries of root basins in Newton's root approximation method. The gr...