protein structural flexibility is important for catalysis, binding, protein design, and allostery. some simple methods have recently been introduced to compute protein flexibility directly from the protein structure without any mechanical models. for example the atomic mean square displacement (or b-factor) is related to the number of neighboring atoms. the protein structure can be modeled as a...

A perfect path double cover (PPDC) of a graph G is a family P of paths of G such that every edge of G belongs to exactly two paths of P and each vertex of G occurs exactly twice as an endpoint of a path in P. Li (J. Graph Theory 14 (1990) 645–650) has shown that every simple graph has a PPDC.A regular perfect path double cover (RPPDC) of a graph G is a PPDC of G in which all paths are of the sa...

An r-simple k-path is a path in the graph of length k that passes through each vertex at most r times. The r-SIMPLE k-PATH problem, given a graph G as input, asks whether there exists an r-simple k-path in G. We first show that this problem is NP-Complete. We then show that there is a graph G that contains an r-simple k-path and no simple path of length greater than 4 log k/ log r. So this, in ...

In this research paper, we present a novel frame work for handling $m$-polar information by combining the theory of $m-$polar fuzzy  sets with graphs. We introduce certain types of edge regular $m-$polar fuzzy graphs and edge irregular $m-$polar fuzzy graphs. We describe some useful properties of edge regular, strongly edge irregular and strongly edge totally irregular $m-$polar fuzzy graphs. W...

It is shown that the existence of a Hamiltonian path in a planar automatic graph of bounded degree is complete for Σ1 1 , the first level of the analytical hierarchy. This sharpens a corresponding result of Hirst and Harel for highly recursive graphs. Furthermore, we also show: (i) The Hamiltonian path problem for finite planar graphs that are succinctly encoded by an automatic presentation is ...

The principal ratio of a connected graph, denoted γ(G), is the ratio of the maximum and minimum entries of its first eigenvector. Cioabă and Gregory conjectured that the graph on n vertices maximizing γ(G) is a kite graph: a complete graph with a pendant path. In this paper we prove their conjecture.

Let A and B be two disjoint sets of points in the plane such that no three points of A ∪ B are collinear, and let n be the number of points in A. A geometric complete bipartite graph K(A, B) is a complete bipartite graph with partite sets A and B which is drawn in the plane such that each edge of K(A, B) is a straight-line segment. We prove that (i) If |B| ≥ (n + 1)(2n − 4) + 1, then the geomet...

Let G be a connected simple (molecular) graph. The distance d(u, v) between two vertices u and v of G is equal to the length of a shortest path that connects u and v. In this paper we compute some distance based topological indices of H-Phenylenic nanotorus. At first we obtain an exact formula for the Wiener index. As application we calculate the Schultz index and modified Schultz index of this...