9. Notation and an outline. Suppose that we have n i.i.d. random matrices Xn = (X(1), X(2), . . . , X(n)) where X(i) ∼ Nf,m (0, A0 ⊗ B0) for all i, and A0 = (ajk) and B0 = (bjk) are positive definite. Let Y (t) = X(t) T , Ŝn = 1 n ∑n t=1 vec {X(t) } vec {X(t) } T , and S̃n = 1 n ∑n t=1 vec {Y(t) } vec {Y(t) } . Denote the `, kth block of size f × f in Ŝn by Ŝ`k n = (Ŝ`k ij ) and that of size m ×...

let $r$ be a ring with unity. the undirected nilpotent graph of $r$, denoted by $gamma_n(r)$, is a graph with vertex set ~$z_n(r)^* = {0neq x in r | xy in n(r) for some y in r^*}$, and two distinct vertices $x$ and $y$ are adjacent if and only if $xy in n(r)$, or equivalently, $yx in n(r)$, where $n(r)$ denoted the nilpotent elements of $r$. recently, it has been proved that if $r$ is a left ar...

given a non-abelian finite group $g$, let $pi(g)$ denote the set of prime divisors of the order of $g$ and denote by $z(g)$ the center of $g$. thetextit{ prime graph} of $g$ is the graph with vertex set $pi(g)$ where two distinct primes $p$ and $q$ are joined by an edge if and only if $g$ contains an element of order $pq$ and the textit{non-commuting graph} of $g$ is the graph with the vertex s...

Recently there has been significant progress in classifying integer friezes and S mathvariant="normal">L 2 </mm...

An irregularity index IR(Γ) of a graph Γ is nonnegative numeric quantity (i.e., IR(Γ)≥0) such that IR(Γ)=0 iff regular graph. In this paper, we show IRC closely correlates with the normal boiling point Tbp and standard heat formation ΔHfo lower benzenoid hydrocarbons. The correlation models fit data efficiently for both are linear. We develop further mathematical properties by calculating its e...

We generalize the notion of a graph automatic group introduced by Kharlampovich, Khoussainov and Miasnikov by replacing the regular languages in their definition with more powerful language classes. For a fixed language class C, we call the resulting groups C-graph automatic. We prove that the class of C-graph automatic groups is closed under change of generating set, direct and free product fo...

The general form of safe recursion (or ramified recurrence) can be expressed by an infinite graph rewrite system including unfolding graph rewrite rules introduced by Dal Lago, Martini and Zorzi, in which the size of every normal form by innermost rewriting is polynomially bounded. Every unfolding graph rewrite rule is precedence terminating in the sense of Middeldorp, Ohsaki and Zantema. Altho...

let $g$ be a non-abelian finite group. in this paper, we prove that $gamma(g)$ is $k_4$-free if and only if $g cong a times p$, where $a$ is an abelian group, $p$ is a $2$-group and $g/z(g) cong mathbb{ z}_2 times mathbb{z}_2$. also, we show that $gamma(g)$ is $k_{1,3}$-free if and only if $g cong {mathbb{s}}_3,~d_8$ or $q_8$.