A signed graph is a graph where the edges are assigned either positive ornegative signs. Net degree of a signed graph is the dierence between the number ofpositive and negative edges incident with a vertex. It is said to be net-regular if all itsvertices have the same net-degree. Laplacian energy of a signed graph is defined asε(L(Σ)) =|γ_1-(2m)/n|+...+|γ_n-(2m)/n| where γ_1,...,γ_n are the ei...

In this paper, we continue the study of the domination game in graphs introduced by Bre{v{s}}ar, Klav{v{z}}ar, and Rall. We study the paired-domination version of the domination game which adds a matching dimension to the game. This game is played on a graph $G$ by two players, named Dominator and Pairer. They alternately take turns choosing vertices of $G$ such that each vertex chosen by Domin...

number of ordered triples (a, b, c) such that ab ∈ E(G) and bc / ∈ E(G). Observe that each vertex v has deg v = 0, 1, 2, 3, 4, or 5. If deg v is 0 or 5, then v is the second vertex in 0 such triples. If deg v is 1 or 4, then v is the second vertex in 4 such triples. If deg v is 2 or 3, then v is the second vertex in 6 such triples. Thus, there are at most 36 such ordered triples. Note that each...

The harmonic index H(G) , of a graph G is defined as the sum of weights 2/(deg(u)+deg(v)) of all edges in E(G), where deg (u) denotes the degree of a vertex u in V(G). In this paper we define the harmonic polynomial of G. We present explicit formula for the values of harmonic polynomial for several families of specific graphs and we find the lower and upper bound for harmonic index in Caterpill...

todeschini et al. have recently suggested to consider multiplicative variants of additive graphinvariants, which applied to the zagreb indices would lead to the multiplicative zagrebindices of a graph g, denoted by ( ) 1  g and ( ) 2  g , under the name first and secondmultiplicative zagreb index, respectively. these are define as  ( )21 ( ) ( )v v gg g d vand ( ) ( ) ( )( )2 g d v d v gu...

For k ∈ N, Corrádi and Hajnal proved that every graph G on 3k vertices with minimum degree δ(G) ≥ 2k has a C3-factor, i.e., a partitioning of the vertex set so that each part induces the 3-cycle C3. Wang proved that every directed graph − → G on 3k vertices with minimum total degree δt( − → G) := minv∈V (deg −(v)+deg+(v)) ≥ 3(3k−1)/2 has a − → C 3-factor, where − → C 3 is the directed 3-cycle. ...

An extended star is a tree which has only one vertex with degree larger than two. The -center problem in a graph  asks to find a subset  of the vertices of  of cardinality  such that the maximum weighted distances from  to all vertices is minimized. In this paper we consider the -center problem on the unweighted extended stars, and present some properties to find solution.