The vertex-edge Wiener polynomials of a simple connected graph are defined based on the distances between vertices and edges of that graph. The first derivative of these polynomials at one are called the vertex-edge Wiener indices. In this paper, we express some basic properties of the first and second vertex-edge Wiener polynomials of simple connected graphs and compare the first and second ve...

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:...

the k-th semi total point graph of a graph g, , ‎is a graph‎ obtained from g by adding k vertices corresponding to each edge and‎ connecting them to the endpoints of edge considered‎. ‎in this paper‎, a formula for laplacian polynomial of in terms of‎ characteristic and laplacian polynomials of g is computed‎, ‎where is a connected regular graph‎.the kirchhoff index of is also computed‎.

The cover polynomial introduced by Chung and Graham is a twovariate graph polynomial for directed graphs. It counts the (weighted) number of ways to cover a graph with disjoint directed cycles and paths, it is an interpolation between determinant and permanent, and it is believed to be a directed analogue of the Tutte polynomial. Jaeger, Vertigan, and Welsh showed that the Tutte polynomial is #...

We investigate the effect of certain natural connectivity constraints on the parameterized complexity of two fundamental graph covering problems, namely k-VERTEX COVER and k-EDGE COVER. Specifically, we impose the additional requirement that each connected component of a solution have at least t vertices (resp. edges from the solution), and call the problem t-TOTAL VERTEX COVER (resp. t-TOTAL E...

The weighted matroid parity problems for the matching matroid and gammoids are among the very few cases for which the weighted matroid parity problem is polynomial time solvable. In this work we extend these problems to a general revenue function for each pair, and show that the resulting problem is still solvable in polynomial time via a standard weighted matching algorithm. We show that in ma...

We consider the stochastic graph model where the location of each vertex is a random point in a given metric space. We study the problems of computing the expected lengths of the minimum spanning tree, the minimum perfect matching and the minimum cycle cover on such a stochastic graph and obtain an FPRAS (Fully Polynomial Randomized Approximation Scheme) for each of these problems. Our result f...

An n-multigraph G = (V ;Ei | i ∈ I) is a complete graph G = (V,E) whose edges are covered by n = |I| sets E = ∪i∈IEi, some of which might be empty. If this cover is a partition, then we call G an n-graph. We say that an n-graph G′ = (V ;E′ i | i ∈ I) is an edge-subgraph of an n-multigraph G = (V ;Ei | i ∈ I) if E′ i ⊆ Ei for all i ∈ I. We denote by ∆ the n-graph on three vertices with three non...

In this paper we study the edge-thickness and the clique-thickness of a graph. The edge-thickness of a graph is defined as the thickness of the family of edges. The clique-thickness of a graph is defined as the thickness of the family of maximal cliques. Edges and maximal cliques of a graph are both considered as a collection of subsets of the vertex set. On one hand, we introduce a new paramet...