Abstract The number of spanning trees of a graph G is the total number of distinct spanning subgraphs of G that are trees (tree that visiting all the vertices of the graph G). Let Cn be a cycle with n vertices. The Star flower planar map is a simple graph G formed from a cycle Cn by adding a vertex adjacent to every edge of Cn and we connect this vertex with two end vertices of each edge of Cn,...

The commuting graph of a finite non-abelian group G with center Z(G), denoted by ?c(G), is simple undirected whose vertex set G?Z(G), and two distinct vertices x y are adjacent if only xy=yx. Alwardi et al. (Bulletin, 2011, 36, 49-59) defined the common neighborhood matrix CN(G) energy Ecn(G) G. A called CN-hyperenergetic Ecn(G)>Ecn(Kn), where n=|V(G)| Kn denotes complete on n vertices. Two ...

A subset of vertices a graph is minimal if, within all subsets the same size, its vertex boundary minimal. We give complete, geometric characterization sets for planar integer lattice $X$. Our elucidates structure sets, and we are able to use it obtain several applications. show that neighborhood set characterize uniquely $X$: those which congruent any other size. also classify efficient have m...

Let D be a finite simple directed graph with vertex set V(D) and arc set A(D). A function ( ) { } − : 1,1 f V D → is called a signed dominating function (SDF) if [ ] ( ) 1 D f N v − ≥ for each vertex v V ∈ . The weight ( ) f ω of f is defined by ( ) ∑ v V f v ∈ . The signed domination number of a digraph D is ( ) ( ) { } γ ω = min is an SDF of s D f f D . Let Cm × Cn denotes the cartesian produ...

A vertex-magic group edge labeling of a graph G(V,E) with |E| = n is an injection from E to an abelian group Γ of order n such that the sum of labels of all incident edges of every vertex x ∈ V is equal to the same element μ ∈ Γ. We completely characterize all Cartesian products Cn Cm that admit a vertex-magic group edge labeling by Z2nm, as well as provide labelings by a few other finite abeli...

The cyclic ladder graph CLn is the Cartesian product of cycles Cn and paths P2, that CLn=Cn×P2, (n≥3). di-forcing polynomial a binary enumerative all perfect matching forcing anti-forcing numbers. In this paper, we derive recursive formulas for by classifying counting cases associated edges given vertex, from which obtain number matching, polynomials, generating function computing some polynomi...