The eccentric distance sum (EDS) is a novel graph invariant which can be used to predict biological and physical properties, and has a vast potential in structure activity/property relationships. For a connected graph G, its EDS is defined as ξ d (G) = ∑ v∈V (G) ecc G (v)D G (v), where ecc G (v) is the eccentricity of a vertex v in G and D G (v) is the sum of distances of all vertices in G from...

For a positive integer k, a total {k}-dominating function of a graph G without isolated vertices is a function f from the vertex set V (G) to the set {0, 1, 2, . . . , k} such that for any vertex v ∈ V (G), the condition ∑ u∈N(v) f(u) ≥ k is fulfilled, where N(v) is the open neighborhood of v. The weight of a total {k}-dominating function f is the value ω(f) = ∑ v∈V f(v). The total {k}-dominati...

We consider the problem of securing a graph against a sequence of vertex attacks by placing guards on its vertices. The guards are “mobile” in that, after any attack, each guard can move to any neighbor of its current location, but one guard must move to the attacked vertex. We determine sharp upper bounds, in terms of the order of the graph, on the minimum number of guards necessary for connec...

Let G be a finite and simple graph with vertex set V (G), and let f : V (G) → {−1, 1} be a two-valued function. If ∑ x∈N [v] f(x) ≤ 1 for each v ∈ V (G), where N [v] is the closed neighborhood of v, then f is a signed 2independence function onG. The weight of a signed 2-independence function f is w(f) = ∑ v∈V (G) f(v). The maximum of weights w(f), taken over all signed 2-independence functions ...

Let G = (V,E) be a graph. A set S ⊆ V is a total restrained dominating set if every vertex is adjacent to a vertex in S and every vertex of V − S is adjacent to a vertex in V − S. A set S ⊆ V is a restrained dominating set if every vertex in V − S is adjacent to a vertex in S and to a vertex in V − S. The total restrained domination number of G (restrained domination number of G, respectively),...

In 1993 Hong asked what are the best bounds on the k’th largest eigenvalue λk(G) of a graph G of order n. This challenging question has never been tackled for any 2 < k < n. In the present paper tight bounds are obtained for all k > 2, and even tighter bounds are obtained for the k’th largest singular value λk(G). Some of these bounds are based on Taylor’s strongly regular graphs, and other on ...

We introduce a notion of vertex association and consider sequences of these associations. This allows for slick proofs of a few known theorems as well as showing that for any induced subgraph H of G, χ(G) ≤ χ(H) + 12 (ω(G) + |G| − |H| − 1). As a special case of this, we have χ(G) ≤ ⌈ ω(G)+τ(G) 2 ⌉ (here χ(G) denotes the chromatic number, ω(G) the clique number and τ(G) the vertex cover number),...

A set S of vertices in a graph G=(V,E) is a dominating set ofG if every vertex of V-S is adjacent to some vertex of S.For an integer k≥1, a set S of vertices is a k-step dominating set if any vertex of $G$ is at distance k from somevertex of S. In this paper, using membership values of vertices and edges in fuzzy graphs, we introduce the concepts of strength of strongestdominating set as well a...

The chromatic number χ(G) of a graph G is the minimum number of colours required to colour the vertices of G in such a way that no two adjacent vertices of G receive the same colour. A partition of V into χ(G) independent sets (called colour classes) is said to be a χpartition of G. A set S ⊆ V is called a dominating set of G if every vertex in V − S is adjacent to a vertex in S. A dominating s...

The economy-climate interaction and an appropriate mitigation policy for climate protection have been treated in various types of scientific modeling. Here, we specifically focus on the seminal work by Nordhaus [14, 15] link. We extend type model to include optimal policies mitigation, adaptation infrastructure investment studying dynamics transition a low fossil-fuel economy. Formally, gives r...