On the number of edges in graphs with a given connected domination number

On the super domination number of graphs

The open neighborhood of a vertex $v$ of a graph $G$ is the set $N(v)$ consisting of all vertices adjacent to $v$ in $G$. For $Dsubseteq V(G)$, we define $overline{D}=V(G)setminus D$. A set $Dsubseteq V(G)$ is called a super dominating set of $G$ if for every vertex $uin overline{D}$, there exists $vin D$ such that $N(v)cap overline{D}={u}$. The super domination number of $G$ is the minimum car...

The Asymptotic Number of Labeled Connected Graphs with a Given Number of Vertices and Edges

Let c(n, q) be the number of connected labeled graphs with n vertices and q 5 N = ( ) edges. Let x = q/n and k = q n. We determine functions w k 1 , a(x ) and cp(x) such that c (n , q) w k ( z ) e n r p ( x ) e o ( x ) uniformly for all n and q 2 n. If Q > O is fixed, n+= and 4q > ( 1 + ~ ) n log n, this formula simplifies to c(n, q) ( t ) exp(-ne-zq’n). On the other hand, if k = o(n”’), this f...

The edge domination number of connected graphs

A subset X of edges in a graph G is called an edge dominating set of G if every edge not in X is adjacent to some edge in X. The edge domination number γ′(G) of G is the minimum cardinality taken over all edge dominating sets of G. Let m,n and k be positive integers with n − 1 ≤ m ≤ (n 2 ) , G(m,n) be the set of all non-isomorphic connected graphs of order n and size m, and G(m,n; k) = {G ∈ G(m...

Preliminaries on the Asymptotic Number of H-free Labeled Connected Graphs with a given Number of Vertices and Edges

On the Number of Certain Subgraphs Contained in Graphs with a given Number of Edges

All graphs considered are finite, undirected, with no loops, no multiple edges and no isolated vertices. For two graphs G, H, let N(G, H) denote the number of subgraphs of G isomorphic to H. Define also, for l >=0, N(I,H)= max N(G, H), where the maximum is taken over all graphs G with l edges. We determine N(l, H) precisely for all l -> 0 when H is a disjoint union of two stars, and also when H...