Let $G$ be a simple undirected graph with each vertex colored
 either white or black, $ u black of G, and
 exactly one neighbor v white. Then change the
 color to black. When this rule is applied, we say $
 forces v, and write \rightarrow $. A $zero\ forcing\ set$ G$ a
 subset $Z$ vertices such that if initially the in Z are remaining vertices
 colored white, entire G may black
 by repeatedly applying color-change rule.
 The zero forcing number G$,...

In this paper the 3-way intersection problem for $S(2,4,v)$ designs is investigated. Let $b_{v}=\frac {v(v-1)}{12}$ and $I_{3}[v]=\{0,1,...,b_{v}\}\setminus\{b_{v}-7,b_{v}-6,b_{v}-5,b_{v}-4,b_{v}-3,b_{v}-2,b_{v}-1\}$. $J_{3}[v]=\{k|$ there exist three with $k$ same common blocks$\}$. We show that $J_{3}[v]\subseteq I_{3}[v]$ any positive integer $v\equiv1, 4\ (\rm mod \ 12)$ $J_{3}[v]=I_{3}[v]$, $ v\geq49$ $v=13 $. find $J_{3}[16]$ completely. Also we determine some values of $J_{3}[v]$ $\...

The zero forcing number $Z(G)$ of a graph $G$ is the minimum cardinality set $S$ with colored (black) vertices which forces $V(G)$ to be after some times. "color change rule": white vertex changed black when it only neighbor vertex. In this case, we say that We investigate here concept connected and number. discusses subject for special graphs products graphs. Also introduce propagation time. Graphs extreme times maximum $|G|-1$ $|G|-2$ are characterized.

A $4-$cycle system is a partition of the edges complete graph $K_n$ into $4-$cycles. Let ${ C}$ be collection cycles length 4 whose $K_n$. set 4-cycles $T_1 \subset C$ called 4-cycle trade if there exists $T_2$ edge-disjoint on same vertices, such that $({C} \setminus T_1)\cup T_2$ also We study trades volume two (double-diamonds) and three show all 4-CS(9) connected with respect trading 2 (double-diamond) 3. In addition, we present full rank matrix null-space containing trade-vectors.

A $μ$-way $(v,k,t)$ $trade$ of volume $m$ consists $μ$ disjoint collections $T_1$, $T_2, \dots T_μ$, each blocks, such that for every $t$-subset $v$-set $V$ the number blocks containing this t-subset is same in $T_i\ (1\leq i\leq μ)$. In other words any pair $\{T_i,T_j\}$, $1\leq i

The investigation of impact fuzzy sets on zero forcing set is the main aim this paper. According to this, results lead us a new concept which we introduce it as Fuzzy Zero Forcing Set (FZFS). We propose and suggest polynomial time algorithm construct FZFS. Further more compute propagation FZFS graphs. This can be efficient model opinion formation problem independent cascade models in social networks. Some examples are provided illustrate constructing special Also, utilize network problem.

The zero forcing number $Z(G)$ of a graph $G$ is the minimum cardinality set $S$ with colored (black) vertices which forces $V(G)$ to be after some times. ``color change rule'': white vertex changed black when it only neighbor vertex. In this case, we say that We investigate here concept connected and number. discusses subject for special graphs products graphs. Also introduce propagation time. Graphs extreme times maximum $|G|-1$ $|G|-2$ are characterized.

A 3-way $(v,k,t)$ trade $T$ of volume $m$ consists three pairwise disjoint collections $T_1$, $T_2$ and $T_3$, each blocks size $k$, such that for every $t$-subset $v$-set $V$, the number containing this is same in $T_i$ $1\leq i\leq 3$. If any found($T$) occurs at most once 3$, then called Steiner trade. We attempt to complete spectrum $S_{3s}(v,k)$, set all possible sizes, $(v,k,2)$ trades, by applying some block designs, as BIBDs, RBs, GDDs, RGDDs, $r\times s$ packing grid blocks....