the crossing number of a graph  is the minimum number of edge crossings over all possible drawings of  in a plane. the crossing number is an important measure of the non-planarity of a graph, with applications in discrete and computational geometry and vlsi circuit design. in this paper we introduce vertex centered crossing number and study the same for maximal planar graph.

‎let $g=(v(g),e(g))$ be a graph‎, ‎$gamma_t(g)$. let $ooir(g)$ be the total domination and oo-irredundance number of $g$‎, ‎respectively‎. ‎a total dominating set $s$ of $g$ is called a $textit{total perfect code}$ if every vertex in $v(g)$ is adjacent to exactly one vertex of $s$‎. ‎in this paper‎, ‎we show that if $g$ has a total perfect code‎, ‎then $gamma_t(g)=ooir(g)$‎. ‎as a consequence, ...

flaviviridae are viruses that cause several diseases including dengue fever, japanese encephalitis, murray valley encephalitis, tick-borne encephalitis, west nile encephalitis, yellow fever and hepatitis c virus infection. members of this family have monopartite, linear, single-stranded rna genomes of positive polarity, 9.6-12.3 kb in length. here, we have analyzed the codon usage of 13 species...

the reverse degree distance of a connected graph $g$ is defined in discrete mathematical chemistry as [ r (g)=2(n-1)md-sum_{uin v(g)}d_g(u)d_g(u), ] where $n$, $m$ and $d$ are the number of vertices, the number of edges and the diameter of $g$, respectively, $d_g(u)$ is the degree of vertex $u$, $d_g(u)$ is the sum of distance between vertex $u$ and all other vertices of $g$, and $v(g)$ is the ...

In the present study we evaluated the amount and distribution of genetic variation by using RAPD marker variation of 12 markers loci in three broodstock groups of rainbow trout. A total of 47 polymorphic bands were observed in Iranian strain, average number of bands was 10 and average number of polymorphic bands per primer was 3.92. The total detected bands in rainbow trout strain originated fr...

The effects of different concentrations of Zinc oxide nano particles (ZnONP; 0, 10, 25, 50, 75 and 100 mg L-1) and Humic acid (HA; 0, 50, 100, 500 and 1000 mg L-1) on root length and number, leaf length and number, bulb diameter, and chlorophyll, total phenol, anthocyanin and flavonoid contents were examined under in vitro conditions. Total phenol, flavonoid and anthocyanin contents were signif...

In the present study we evaluated the amount and distribution of genetic variation by using RAPD marker variation of 12 markers loci in three broodstock groups of rainbow trout. A total of 47 polymorphic bands were observed in Iranian strain, average number of bands was 10 and average number of polymorphic bands per primer was 3.92. The total detected bands in rainbow trout strain originated fr...

a {em 2-rainbow dominating function} (2rdf) of a graph $g$ is a function $f$ from the vertex set $v(g)$ to the set of all subsets of the set ${1,2}$ such that for any vertex $vin v(g)$ with $f(v)=emptyset$ the condition $bigcup_{uin n(v)}f(u)={1,2}$ is fulfilled, where $n(v)$ is the open neighborhood of $v$. the {em weight} of a 2rdf $f$ is the value $omega(f)=sum_{vin v}|f (v)|$. the {em $2$-r...