The prime graph of a finite group $G$ is denoted by $Gamma(G)$ whose vertex set is $pi(G)$ and two distinct primes $p$ and $q$ are adjacent in $Gamma(G)$, whenever $G$ contains an element with order $pq$. We say that $G$ is unrecognizable by prime graph if there is a finite group $H$ with $Gamma(H)=Gamma(G)$, in while $Hnotcong G$. In this paper, we consider finite groups with the same prime gr...

OBJECTIVE To construct monochorionic diamniotic (MCDA) and dichorionic diamniotic (DCDA) twin reference ranges for umbilical artery (UA) pulsatility index (PI), UA resistance index (RI), fetal middle cerebral artery (MCA) PI and peak systolic velocity (PSV) and cerebroplacental ratio (CPR) from 24 weeks' to 38 weeks' gestation and compare these with published normal values for singleton pregnan...

BACKGROUND In species across taxa, offspring have means to influence parental investment (PI). PI thus evolves as an interacting phenotype and indirect genetic effects may strongly affect the co-evolutionary dynamics of offspring and parental behaviors. Evolutionary theory focused on explaining how exaggerated offspring solicitation can be understood as resolution of parent-offspring conflict, ...

Since the early 1990s human immunodeficiency virus (HIV)/acquired immunodeficiency syndrome (AIDS) emerged as a global health pandemic, with sub-Saharan Africa the hardest hit. While the successful roll-out of antiretroviral (ARV) therapy provided significant relief to HIV-positive individuals, such treatment can also elicit damaging side-effects. Here especially HIV protease inhibitors (PIs) a...

Let G be a simple graph with vertex set and edge set . The function which assigns to each pair of vertices in , the length of minimal path from to , is called the distance function between two vertices. The distance function between and edge and a vertex is where for and. , . The Wiener index of a graph is denoted by and is defined by .In general this kind of index is called a topological index...

Let G= V E be a simple connected graph. The distance between two vertices of G is defined to be the length of the shortest path between the two vertices. There are topological indices assigned to G and based on the distance function which are invariant under the action of the automorphism group of G. Some important indices assigned to G are the Wiener, Szeged, and PI index which we will find th...

In a graph G, a vertex dominates itself and its neighbors. A subset S ⊆ V (G) is a double dominating set of G if S dominates every vertex of G at least twice. The minimum cardinality of a double dominating set of G is the double domination number γ×2(G). A function f(p) is defined, and it is shown that γ×2(G) = min f(p), where the minimum is taken over the n-dimensional cube C = {p = (p1, . . ....

The values of the masses particles involved in decay ${T}_{cc}^{+}\ensuremath{\rightarrow}{D}^{*+}{D}^{0}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{+}{D}^{0}{D}^{0}$ suggest that due to final state interactions transition vertex ${T}_{cc}^{+}\ensuremath{\rightarrow}{D}^{*+}{D}^{0}$ there may be triangle logarithmic singularities. We discuss their possible role and show tree approximation for c...

Consider two parallel lines (denoted r1 and r2). A graph is a PI graph (Point-Interval graph) if it is an intersection graph of a family F of triangles between r1 and r2 such that each triangle has an interval with two endpoints on r1 and a vertex (a point) on r2. The family F is the PI representation of G. The PI graphs are an extension of interval and permutation graphs and they form a subcla...

A pebbling move refers to the act of removing two pebbles from one vertex and placing pebble on an adjacent vertex. The goal graph is: Given initial distribution pebbles, use moves reach a specified called root. number \(\pi (G)\) is minimum needed so every can choice We introduce new variant pebbling, game between players. One player aims root other prevent this. show configurations various cl...