If G is a connected graph with vertex set V , then the eccentric connectivity index of G, ξ(G) is defined as ∑ deg(v).ec(v) where deg(v) is the degree of a vertex v and ec(v) is its eccentricity. The Wiener index W (G) = 1 2 [ ∑ d(u, v)], the hyper-Wiener index WW (G) = 1 2 [ ∑ d(u, v) + ∑ d(u, v)] and the reverseWiener index ∧(G) = n(n−1)D 2 −W (G), where d(u, v) is the distance of two vertice...

the prime graph $gamma(g)$ of a group $g$ is a graph with vertex set $pi(g)$, the set of primes dividing the order of $g$, and two distinct vertices $p$ and $q$ are adjacent by an edge written $psim q$ if there is an element in $g$ of order $pq$. let $pi(g)={p_{1},p_{2},...,p_{k}}$. for $pinpi(g)$, set $deg(p):=|{q inpi(g)| psim q}|$, which is called the degree of $p$. we also set $d(g):...

Controlled cell division is central to the growth and development of all multicellular organisms. Within the proliferating zone of the Arabidopsis root, regular symmetric divisions give rise to patterns of parallel files of cells, the genetic basis of which remains unclear. We found that genotypes impaired in the TONNEAU1a (TON1a) gene display misoriented symmetric divisions in the epidermis an...

The Randić index R(G) of a graph G is the sum of weights (deg(u) deg(v))−0.5 over all edges uv of G, where deg(v) denotes the degree of a vertex v. Let r(G) be the radius of G. We prove that for any connected graph G of maximum degree four which is not a path with even number of vertices, R(G) ≥ r(G). As a consequence, we resolve the conjecture R(G) ≥ r(G)− 1 given by Fajtlowicz in 1988 for the...

We consider topological indices I that are sums of f(deg(u))f(deg(v)), where {u, v} are adjacent vertices and f is a function. The Randić connectivity index or the Zagreb group index are examples for indices of this kind. In earlier work on topological indices that are sums of independent random variables, we identified the correlation between I and the edge set of the molecular graph as the ma...

The Wiener indexW (G) of a connected graphG is defined to be the sum ∑ u,v d(u, v) of the distances between the pairs of vertices in G. Similarly, the edge-Wiener index We(G) of G is defined to be the sum ∑ e,f d(e, f) of the distances between the pairs of edges in G, or equivalently, the Wiener index of the line graph L(G). Finally, the Gutman index Gut(G) is defined to be the sum ∑ u,v deg(u)...

This is the first study to demonstrate the selectivity of learning for contrast polarity. The finding is the main result of an investigation into the existence of central and peripheral vision mechanisms selective for contrast polarity within the texture-segregation process, using the perceptual learning paradigm in a detection task. Energy models (Malik & Perona, 1990) exclude segregation of t...

Stem cells (SCs) perform the task of maintaining tissue homeostasis by both self-renewal and differentiation. While it has been argued that SCs divide asymmetrically, there is also evidence that SCs undergo symmetric division. Symmetric SC division has been speculated to be key for expanding cell numbers in development and regeneration after injury. However, it might lead to uncontrolled growth...