A lower bound for the independence number of a graph is a graph invariant l such that, for every graph G, l(G) ≤ α(G). Similarly, an upper bound for the independence number is a graph invariant u such that, for every graph G, α(G) ≤ u(G). Many efficiently computable upper and lower bounds, called α-bounds here, have been published and these are surveyed in the following section. They can be use...

Let G be a simple undirected graph. A broadcast on G is a function f : V (G) → N such that f(v) ≤ eG(v) holds for every vertex v of G, where eG(v) denotes the eccentricity of v in G, that is, the maximum distance from v to any other vertex of G. The cost of f is the value cost(f) = ∑ v∈V (G) f(v). A broadcast f on G is independent if for every two distinct vertices u and v in G, dG(u, v) > max{...

The concept of inverse domination was introduced by Kulli V.R. and Sigarakanti S.C. [9] . Let D be a  set of G. A dominating set D1  VD is called an inverse dominating set of G with respect to D. The inverse domination number   (G) is the order of a smallest inverse dominating set. Motivated by this definition we define another parameter as follows. Let D be a maximum independent set in G. ...

Let K (3) 4 denote the complete 3-uniform hypergraph on 4 vertices. Ajtai, Erdős, Komlós, and Szemerédi (1981) asked if there is a function ω(d) → ∞ such that every 3-uniform, K (3) 4 -free hypergraph H with N vertices and average degree d has independence number at least N d1/2 ω(d). We answer this question by constructing a 3-uniform, K (3) 4 -free hypergraph with independence number at most ...

In this paper we will define a cardinal invariant corresponding to the independence number for partitions of ω. By using Cohen forcing we will prove that this cardinal invariant is consistently smaller than the continuum.

The independence number of a graph is defined as the maximum size of a set of pairwise non-adjacent vertices and the spectral radius is defined as the maximum eigenvalue of the adjacency matrix of the graph. Xu et al. in [The minimum spectral radius of graphs with a given independence number, Linear Algebra and its Applications 431 (2009) 937–945] determined the connected graphs of order n with...