we define minimal cn-dominating graph $mathbf {mcn}(g)$, commonality minimal cn-dominating graph $mathbf {cmcn}(g)$ and vertex minimal cn-dominating graph $mathbf {m_{v}cn}(g)$, characterizations are given for graph $g$ for which the newly defined graphs are connected. further serval new results are developed relating to these graphs.

a vertex irregular total k-labeling of a graph g with vertex set v and edge set e is an assignment of positive integer labels {1, 2, ..., k} to both vertices and edges so that the weights calculated at vertices are distinct. the total vertex irregularity strength of g, denoted by tvs(g)is the minimum value of the largest label k over all such irregular assignment. in this paper, we study the to...

A vertex irregular total k-labeling of a graph G with vertex set V and edge set E is an assignment of positive integer labels {1, 2, ..., k} to both vertices and edges so that the weights calculated at vertices are distinct. The total vertex irregularity strength of G, denoted by tvs(G)is the minimum value of the largest label k over all such irregular assignment. In this paper, we study the to...

in this paper, we define the common minimal common neighborhooddominating signed graph (or common minimal $cn$-dominating signedgraph) of a given signed graph and offer a structuralcharacterization of common minimal $cn$-dominating signed graphs.in the sequel, we also obtained switching equivalencecharacterization: $overline{sigma} sim cmcn(sigma)$, where$overline{sigma}$ and $cmcn(sigma)$ are ...

let z2 = {0, 1} and g = (v ,e) be a graph. a labeling f : v → z2 induces an edge labeling f* : e →z2 defined by f*(uv) = f(u).f (v). for i ε z2 let vf (i) = v(i) = card{v ε v : f(v) = i} and ef (i) = e(i) = {e ε e : f*(e) = i}. a labeling f is said to be vertex-friendly if | v(0) − v(1) |≤ 1. the vertex balance index set is defined by {| ef (0) − ef (1) | : f is vertex-friendly}. in this paper ...

Let Z2 = {0, 1} and G = (V ,E) be a graph. A labeling f : V → Z2 induces an edge labeling f* : E →Z2 defined by f*(uv) = f(u).f (v). For i ε Z2 let vf (i) = v(i) = card{v ε V : f(v) = i} and ef (i) = e(i) = {e ε E : f*(e) = i}. A labeling f is said to be Vertex-friendly if | v(0) − v(1) |≤ 1. The vertex balance index set is defined by {| ef (0) − ef (1) | : f is vertex-friendly}. In this paper ...

Let G = (V,E) be a simple graph. A set S ⊆ V is a dominating set of G, if every vertex in V \S is adjacent to at least one vertex in S. Let C n be the family of dominating sets of a cycle Cn with cardinality i, and let d(Cn, i) = |C i n|. In this paper, we construct C i n, and obtain a recursive formula for d(Cn, i). Using this recursive formula, we consider the polynomial D(Cn, x) = ∑n i=⌈n 3 ...

A vertex v in an exponential dominating set assigns weight ( 1 2 )dist(v,u) to vertex u. An exponential dominating set of a graph G is a subset of V (G) such that every vertex in V (G) has been assigned a sum weight of at least 1. In this paper the exponential dominating number for the graph Cm × Cn, denoted by γe(Cm × Cn) is discussed. Anderson et. al. [1] proved that mn 15.875 ≤ γe(Cm × Cn) ≤...

A graph G is called (H; k)-vertex stable if G contains a subgraph isomorphic to H ever after removing any k of its vertices; stab(H; k) denotes the minimum size among the sizes of all (H; k)-vertex stable graphs. In this paper we deal with (Cn; k)vertex stable graphs with minimum size. For each n we prove that stab(Cn; 1) is one of only two possible values and we give the exact value for infini...

To every vertex algebra V we associate a canonical decreasing sequence of subspaces and prove that the associated graded vector space gr(V ) is naturally a vertex Poisson algebra, in particular a commutative vertex algebra. We establish a relation between this decreasing sequence and the sequence Cn introduced by Zhu. By using the (classical) algebra gr(V ), we prove that for any vertex algebra...