let $a$ be a non-trivial abelian group and $a^{*}=asetminus {0}$. a graph $g$ is said to be $a$-magic graph if there exists a labeling$l:e(g)rightarrow a^{*}$ such that the induced vertex labeling$l^{+}:v(g)rightarrow a$, define by $$l^+(v)=sum_{uvin e(g)} l(uv)$$ is a constant map.the set of all constant integerssuch that $sum_{uin n(v)} l(uv)=c$, for each $vin n(v)$,where $n(v)$ denotes the s...

the first zagreb index, $m_1(g)$, and second zagreb index, $m_2(g)$, of the graph $g$ is defined as $m_{1}(g)=sum_{vin v(g)}d^{2}(v)$ and $m_{2}(g)=sum_{e=uvin e(g)}d(u)d(v),$ where $d(u)$ denotes the degree of vertex $u$. in this paper, the firstand second maximum values of the first and second zagreb indicesin the class of all $n-$vertex tetracyclic graphs are presented.

An equitable k-coloring of a graph is defined by a partition of its vertices into k disjoint stable subsets, such that the difference between the cardinalities of any two subsets is at most one. The equitable coloring problem consists of finding the minimum value of k such that a given graph can be equitably k-colored. We present two new integer programming formulations based on representatives...

An L(2, 1)-labeling of a graph G is equitable if the number of elements in any two color classes differ by at most one. The equitable L(2, 1)-labeling number λe(G) of G is the smallest integer k such that G has an equitable L(2, 1)-labeling. Sierpiński graphs S(n, k) generalize the Tower of Hanoi graphs—the graph S(n, 3) is isomorphic to the graph of the Tower of Hanoi with n disks. In this pap...

Let $G$ be a graph with vertex set $V(G)$ and edge set $E(G)$, and let $d_u$ denote the degree of vertex $u$ in $G$. The Randi'c index of $G$ is defined as${R}(G) =sum_{uvin E(G)} 1/sqrt{d_ud_v}.$In this paper, we investigate the relationships between Randi'cindex and several topological indices.

An outer-independent double Italian dominating function (OIDIDF)on a graph $G$ with vertex set $V(G)$ is a function$f:V(G)longrightarrow {0,1,2,3}$ such that if $f(v)in{0,1}$ for a vertex $vin V(G)$ then $sum_{uin N[v]}f(u)geq3$,and the set $ {uin V(G)|f(u)=0}$ is independent. The weight ofan OIDIDF $f$ is the value $w(f)=sum_{vin V(G)}f(v)$. Theminimum weight of an OIDIDF on a graph $G$ is cal...

A total Roman dominating function on a graph $G$ is a function $f: V(G) rightarrow {0,1,2}$ such that for every vertex $vin V(G)$ with $f(v)=0$ there exists a vertex $uin V(G)$ adjacent to $v$ with $f(u)=2$, and the subgraph induced by the set ${xin V(G): f(x)geq 1}$ has no isolated vertices. The total Roman domination number of $G$, denoted $gamma_{tR}(G)$, is the minimum weight $omega(f)=sum_...

In this paper we introduce remainder cordial labeling of graphs. Let $G$ be a $(p,q)$ graph. Let $f:V(G)rightarrow {1,2,...,p}$ be a $1-1$ map. For each edge $uv$ assign the label $r$ where $r$ is the remainder when $f(u)$ is divided by $f(v)$ or $f(v)$ is divided by $f(u)$ according as $f(u)geq f(v)$ or $f(v)geq f(u)$. The function$f$ is called a remainder cordial labeling of $G$ if $left| e_{...

The notion of graph covers is a discretization covering spaces introduced and deeply studied in topology. In discrete mathematics theoretical computer science, they have attained lot attention from both the structural complexity perspectives. Nonetheless, disconnected graphs were usually omitted considerations with explanation that it sufficient to understand coverings connected components targ...