It was conjectured by Thomassen ([B. Alspach, C. Godsil, Cycle in graphs, Ann. Discrete Math. 27 (1985)], p. 466) that every longest circuit of a 3-connected graph must have a chord. This conjecture is verified for locally 4-connected planar graphs, that is, let N be the set of natural numbers; then there is a function h : N → N such that, for every 4-connected graph G embedded in a surface S w...

Let Tn denote the set of all unrooted and unlabeled trees with n vertices, and (i, j) a double-star. By assuming that every tree of Tn is equally likely, we show that the limiting distribution of the number of occurrences of the double-star (i, j) in Tn is normal. Based on this result, we obtain the asymptotic value of Randić index for trees. Fajtlowicz conjectured that for any connected graph ...

A functional connectivity MRI method that groups neighboring voxels in relation to their degree of temporal cross-correlation between their time courses is presented. This grouping generates a vector field, which is assumed to provide insights into the local organization of neuronal activity. Application with high spatial resolution fMRI rat data subjected to electric forepaw sensory stimulatio...

For a 2-connected matroid M , Cunningham and Edmonds gave a tree decomposition that displays all of its 2-separations. When M is 3-connected, two 3-separations are equivalent if one can be obtained from the other by passing through a sequence of 3-separations each of which is obtained from its predecessor by moving a single element from one side of the 3-separation to the other. Oxley, Semple, ...

Let G be a 4-connected graph. For an edge e of G, we do the following operations on G: first, delete the edge e from G, resulting the graph G − e; second, for all the vertices x of degree 3 in G− e, delete x from G− e and then completely connect the 3 neighbors of x by a triangle. If multiple edges occur, we use single edges to replace them. The final resultant graph is denoted by G a e. If G a...

the eccentricity connectivity index of a molecular graph g is defined as (g) = av(g)deg(a)ε(a), where ε(a) is defined as the length of a maximal path connecting a to othervertices of g and deg(a) is degree of vertex a. here, we compute this topological index forsome infinite classes of dendrimer graphs.

This paper presents an architecture and protocol in support of proactive mobility for future IP Wireless Access Networks (IP-WANs). It encompasses a novel approach for seamless handoff and proactive relocation of IP roaming state. The latter establishes a generic substrate for proactive state relocation of different context classes relating to the state of IP connectivity for a mobile node (MN)...

MOTIVATION A variety of biological cellular processes are achieved through a variety of extracellular regulators, signal transduction, protein-protein interactions and differential gene expression. Understanding of the mechanisms underlying these processes requires detailed molecular description of the protein and gene networks involved. To better understand these molecular networks, we propose...

in this paper we present explicit formulas for the eccentric connectivity index of three classesof chain hexagonal cacti. further, it is shown that the extremal chain hexagonal cacti withrespect to the eccentric connectivity index belong to one of the considered types. some openproblems and possible directions of further research are mentioned in the concluding section.

in this paper, we calculate the eccentric connectivity index and the eccentricity sequence of two infinite classes of fullerenes with 50 + 10k and 60 + 12k (k in n) carbon atoms.