THE CONJUGATE FORMAL PRODUCT

Let G be a simple graph of order n and let V (G) be its vertex set. Let c = a + b √ m and c = a − b √ m, where a and b are two nonzero integers and m is a positive integer such that m is not a perfect square. We say that A c = [cij] is the conjugate adjacency matrix of the graph G if cij = c for any two adjacent vertices i and j, cij = c for any two nonadjacent vertices i and j, and cij = 0 if i = j. Let P c G(λ) = |λI − A c | denote the conjugate characteristic polynomial of G and let [A c ij] = {λI −A c}, where {M} denotes the adjoint matrix of a square matrix M. For any two subsets X, Y ⊆ V (G) define 〈X, Y 〉 c = � � A c ij. The expression 〈X, Y 〉 c is called the conjugate i∈X j∈Y formal product of the sets X and Y, associated with the graph G. Using the conjugate formal product we continue our previous investigations of some properties of the conjugate characteristic polynomial of G. 1

Nordhaus-Gaddum results for the convex domination number of a graph

10.1007/s10998-012-2174-7The distance d G (u, v) between two vertices u and v in a connected graph G is the length of the shortest uv-path in G. A uv-path of length d G (u, v) is called a uv-geodesic. A set X is convex in G if vertices from all ab-geodesics belong to X for any two vertices a, b ? X. The convex domination number ?con(G) of a graph G equals the minimum cardinality of a convex dominating set. In the paper, Nordhaus-Gaddum-type results for the convex domination number are studied