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):...

The study of asymptotic minimum degree thresholds that force matchings and tilings in hypergraphs is a lively area research combinatorics. A key breakthrough this was result H\`{a}n, Person Schacht who proved the vertex threshold for perfect matching an $n$-vertex $3$-graph $\left(\frac{5}{9}+o(1)\right)\binom{n}{2}$. In paper we improve on result, giving family sequence results, all which impl...

A n-vertex graph is said to be decomposable if for any partition (λ1, . . . , λp) of the integer n, there exists a sequence (V1, . . . , Vp) of connected vertex-disjoint subgraphs with |Vi| = λi. In this paper, we focus on decomposable trees. We show that a decomposable tree has degree at most 4. Moreover, each degree-4 vertex of a decomposable tree is adjacent to a leaf. This leads to a polyno...

let g be a simple connected graph. the first and second zagreb indices have been introducedas  vv(g)(v)2 m1(g) degg and m2(g)  uve(g)degg(u)degg(v) , respectively,where degg v(degg u) is the degree of vertex v (u) . in this paper, we define a newdistance-based named hyperzagreb as e uv e(g) .(v))2 hm(g)     (degg(u)  degg inthis paper, the hyperzagreb index of the cartesian product...

the non-commuting graph $nabla(g)$ of a non-abelian group $g$ is defined as follows: its vertex set is $g-z(g)$ and two distinct vertices $x$ and $y$ are joined by an edge if and only if the commutator of $x$ and $y$ is not the identity. in this paper we 'll prove that if $g$ is a finite group with $nabla(g)congnabla(bs_{n})$, then $g cong bs_{n}$, where $bs_{n}$ is the symmetric group of degre...