Let k ≥ 2 be an integer, a k-decomposition (G1,G2, . . . ,Gk) of the complete graph Kn is a partition of its edge set to form k spanning subgraphs G1,G2, . . . ,Gk. In this contribution, we investigate the Nordhaus–Gaddum-type inequality of a k-decomposition of Kn for the general Zagreb index and a 2-decomposition for the Zagreb co-indices, respectively. The corresponding extremal graphs are ch...

For a graph G, its k-rainbow independent domination number, written as γrik(G), is defined the cardinality of minimum set consisting k vertex-disjoint sets V1,V2,…,Vk such that every vertex in V0=V(G)\(∪i=1kVi) has neighbor Vi for all i∈{1,2,…,k}. This invariant was proposed by Kraner Šumenjak, Rall and Tepeh (in Applied Mathematics Computation 333(15), 2018: 353–361), which aims to compute num...

We find all maximal linklessly embeddable graphs of order up to 11, and verify that for every graph [Formula: see text] 11 either or its complement is intrinsically linked. give an example a such both are text]-minor free. provide minimal examples not triangular three-connected. prove Nordhaus–Gaddum-type conjecture on the Colin de Verdière invariant at most vertices. description programs used ...

In this paper, we define the notions of inverse strong non-split r-dominating set and inverse strong non-split r-domination number γ′snsr(G) of a graph G. We characterize graphs for which γsnsr(G) + γ′snsr(G) = n, where γsnsr(G) is the strong non-split r-domination number of G. We get many bounds on γ′snsr(G). Nordhaus-Gaddum type results are also obtained for this new parameter.

In this paper the authors study the edge-integrity of graphs. Edge-integrity is a very useful measure of the vulnerability of a network, in particular a communication network, to disruption through the deletion of edges. A number of problems are examined, including some Nordhaus-Gaddum type results. Honest graphs, i.e. those which have the maximum possible edge-integrity, are also investigated....

We consider four conjectures related to the largest eigenvalue of (the adjacency matrix of) a graph (i.e., to the index of the graph). Three of them have been formulated after some experiments with the programming system AutoGraphiX (AGX), designed for finding extremal graphs with respect to given properties by the use of variable neighborhood search. The conjectures are related to the maximal ...

Let G be a connected graph of order n with Laplacian eigenvalues μ1 ≥ μ2 ≥ . . .≥ μn−1 > μn = 0. The Kirchhoff index of G is defined as Kf = Kf(G) = n∑n−1 k=1 1/μk. In this paper. we give lower and upper bounds on Kf of graphs in terms on n, number of edges, maximum degree, and number of spanning trees. Moreover, we present lower and upper bounds on the Nordhaus–Gaddum-type result for the Kirch...

The emph{Harary index} $H(G)$ of a connected graph $G$ is defined as $H(G)=sum_{u,vin V(G)}frac{1}{d_G(u,v)}$ where $d_G(u,v)$ is the distance between vertices $u$ and $v$ of $G$. The Steiner distance in a graph, introduced by Chartrand et al. in 1989, is a natural generalization of the concept of classical graph distance. For a connected graph $G$ of order at least $2$ ...