A Nordhaus-Gaddum problem for a graph parameter is to determine a tight lower or upper bound for the sum or product of the parameter evaluated on a graph and on its complement. This article surveys Nordhaus-Gaddum results for the Colin de Verdière type parameters μ,ν , and ξ ; tree-width and its variants largeur d’arborescence, path-width, and proper path-width; and minor monotone ceilings of v...

A Nordhaus-Gaddum-type result is a (tight) lower or upper bound on the sum or product of a parameter of a graph and its complement. In this paper we study Nordhaus-Gaddum-type results for total domination. We examine the sum and product of γt(G1) and γt(G2) where G1 ⊕G2 = K(s, s), and γt is the total domination number. We show that the maximum value of the sum of the total domination numbers of...

ing and Indexing: Zentralblatt MATH. AUTHORS INFO ARTICLE INFO JOURNAL INFO 2 Nordhaus-Gaddum-type inequalities

A b s t r a c t. Let G denote the complement of the graph G . If I(G) is some invariant of G , then relations (identities, bounds, and similar) pertaining to I(G) + I(G) are said to be of Nordhaus-Gaddum type. A number of lower and upper bounds of Nordhaus-Gaddum type are obtained for the energy and Laplacian energy of graphs. Also some new relations for the Laplacian graph energy are established.

The general Randić index of an organic molecule whose molecular graph is G is defined as the sum of (d(u)d(v))α over all pairs of adjacent vertices of G, where d(u) is the degree of the vertex u in G and α is a real number with α 6= 0. In the paper, we obtained sharp bounds on the general Randić index of the Nordhaus-Gaddum type for trees. Also we show that the general Randić index of the Nordh...

A traditional Nordhaus-Gaddum problem for a graph parameter β is to find a (tight) upper or lower bound on the sum or product of β(G) and β(G) (where G denotes the complement of G). An r-decomposition G1, . . . , Gr of the complete graph Kn is a partition of the edges of Kn among r spanning subgraphs G1, . . . , Gr. A traditional Nordhaus-Gaddum problem can be viewed as the special case for r =...

The distance dG(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 dG(u, v) is called uv-geodesic. A set X is convex in G if vertices from all ab-geodesics belong to X for every two vertices a, b ∈ X. The convex domination number γcon(G) of a graph G equals the minimum cardinality of a convex dominating set. There are a larg...

The energy of a graph G, denoted by E(G), is the sum of the absolute values of all eigenvalues of G . In this paper we present some lower and upper bounds for E(G) in terms of number of vertices, number of edges, and determinant of the adjacency matrix. Our lower bound is better than the classical McClelland’s lower bound. In addition, Nordhaus–Gaddum type results for E(G) are established.

The tight upper bound pt+(G)≤⌈|V(G)|−Z+(G)2⌉ is established for the positive semidefinite propagation time of a graph in terms its zero forcing number. To prove this bound, two methods transforming one set into another and algorithms implementing these are presented. Consequences including Nordhaus-Gaddum sum on time, established.

A dominating set S of graph G is called metric-locating-dominating if it is also locating, that is, if every vertex v is uniquely determined by its vector of distances to the vertices in S . If moreover, every vertex v not in S is also uniquely determined by the set of neighbors of v belonging to S , then it is said to be locating-dominating. Locating, metric-locating-dominating and locatingdom...