In the k disjoint shortest paths problem (k-DSPP), we are given a graph and its vertex pairs (s1, t1), . . . , (sk, tk), and the objective is to find k pairwise disjoint paths P1, . . . , Pk such that each path Pi is a shortest path from si to ti, if they exist. If the length of each edge is equal to zero, then this problem amounts to the disjoint paths problem, which is one of the well-studied...

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....

Let G = (V,E) be an undirected graph, λk be the k th smallest eigenvalue of the normalized laplacian matrix of G. There is a basic fact in algebraic graph theory that λk > 0 if and only if G has at most k − 1 connected components. We prove a robust version of this fact. If λk > 0, then for some 1 ≤ l ≤ k − 1, V can be partitioned into l sets P1, . . . , Pl such that each Pi is a low-conductance...

The Satisfactory Bisection problem means to decide whether a given graph has a partition of its vertex set into two parts of the same cardinality such that each vertex has at least as many neighbors in its part as in the other part. A related variant of this problem, called Co-Satisfactory Bisection, requires that each vertex has at most as many neighbors in its part as in the other part. A ver...

In this paper, at first we introduce a new index with the name Co-PI index and obtain some properties related this new index. Then we compute this new index for TUC4C8(R) nanotubes.

We study a variation of the vertex cover problem where it is required that the graph induced by the vertex cover is connected. We prove that this problem is polynomial in chordal graphs, has a PTAS in planar graphs, is APX-hard in bipartite graphs and is 5/3-approximable in any class of graphs where the vertex cover problem is polynomial (in particular in bipartite graphs). Finally, dealing wit...

We prove polynomial decay of the mixing field Vertex Reinforced Jump Process (VRJP) on ${\mathbb {Z}}^{2}$ with bounded conductances. Using [22] we deduce that VRJP any constant conductances is almost surely recurrent. It gives a counterpart result Merkl, Rolles [16] and Sabot, Zeng for 2-dimensional Edge Random Walk.

Abstract. Proper vertex colorings of a graph are related to its boundary map, also called its signed vertex-edge incidence matrix. The vertex Laplacian of a graph, a natural extension of the boundary map, leads us to introduce nowhere-harmonic colorings and analogues of the chromatic polynomial and Stanley’s theorem relating negative evaluations of the chromatic polynomial to acyclic orientatio...

We study a variation of the vertex cover problem where it is required that the graph induced by the vertex cover is connected. We prove that this problem is polynomial in chordal graphs, has a PTAS in planar graphs, is APX-hard in bipartite graphs and is 5/3-approximable in any class of graphs where the vertex cover problem is polynomial (in particular in bipartite graphs).