in the present study, the effects of two arbuscular mycorrhizal fungi, glomus mosseae andglomus intraradices, alone and in combination, on the growth parameters, chlorophyll and the root-rot disease of pea (pisum sativum) caused by fusarium solani f.sp. pisi were evaluated under greenhouse conditions. eight different treatments in four replications were designed in randomized blocks. compared w...

the first ($pi_1$) and the second $(pi_2$) multiplicative zagreb indices of a connected graph $g$, with vertex set $v(g)$ and edge set $e(g)$, are defined as $pi_1(g) = prod_{u in v(g)} {d_u}^2$ and $pi_2(g) = prod_{uv in e(g)} {d_u}d_{v}$, respectively, where ${d_u}$ denotes the degree of the vertex $u$. in this paper we present a simple approach to order these indices for connected graphs on ...

a function $f:v(g)rightarrow {-1,0,1}$ is a {em minusdominating function} if for every vertex $vin v(g)$, $sum_{uinn[v]}f(u)ge 1$. a minus dominating function $f$ of $g$ is calleda {em global minus dominating function} if $f$ is also a minusdominating function of the complement $overline{g}$ of $g$. the{em global minus domination number} $gamma_{g}^-(g)$ of $g$ isdefined as $gamma_{g}^-(g)=min{...

let g(v,e) be a graph with p vertices and q edges. a graph g is said to have an odd mean labeling if there exists a function f : v (g) → {0, 1, 2,...,2q - 1} satisfying f is 1 - 1 and the induced map f* : e(g) → {1, 3, 5,...,2q - 1} defined by f*(uv) = (f(u) + f(v))/2 if f(u) + f(v) is evenf*(uv) = (f(u) + f(v) + 1)/2 if f(u) + f(v) is odd is a bijection. a graph that admits an odd mean lab...

‎Let G=(V,E) be a simple connected graph with vertex set V and edge set E. The first, second and third Zagreb indices of G are respectivly defined by: $M_1(G)=sum_{uin V} d(u)^2, hspace {.1 cm} M_2(G)=sum_{uvin E} d(u).d(v)$ and $ M_3(G)=sum_{uvin E}| d(u)-d(v)| $ , where d(u) is the degree of vertex u in G and uv is an edge of G connecting the vertices u and v. Recently, the first and second m...

(7) 2 · (p+q) ∈ L(p,q). (8) If p1 = q1 and q1 = (p1)1 and p2 ≤ q2 and q2 ≤ (p1)2, then q ∈ L(p, p1). (9) If p1 ≤ q1 and q1 ≤ (p1)1 and p2 = q2 and q2 = (p1)2, then q ∈ L(p, p1). (10) Let D be a non empty set, given i, j, and M be a matrix over D. If 1 ≤ i and i ≤ lenM and 1 ≤ j and j ≤ widthM, then 〈i, j〉 ∈ the indices of M. (11) If 1 ≤ i and i + 1 ≤ lenG and 1 ≤ j and j + 1 ≤ widthG, then 2 · ...

‎we characterize those groups $g$ and vector spaces $v$ such that $v$ is a faithful irreducible $g$-module and such that each $v$ in $v$ is centralized by a $g$-conjugate of a fixed non-identity element of the fitting subgroup $f(g)$ of $g$‎. ‎we also determine those $v$ and $g$ for which $v$ is a faithful quasi-primitive $g$-module and $f(g)$ has no regular orbit‎. ‎we do use these to show in ...