Average Degree in Graph Powers

Number of walks and degree powers in a graph

This letter deals with the relationship between the total number of k-walks in a graph, and the sum of the k-th powers of its vertex degrees. In particular, it is shown that the sum of all k-walks is upper bounded by the sum of the k-th powers of the degrees. Let G = (V,E) be a connected graph on n vertices, V = {1, 2, . . . , n}, with adjacency matrix A. For any integer k ≥ 1, let a ij denote ...

Every Graph of Sufficiently Large Average Degree Contains a C4-Free Subgraph of Large Average Degree

We prove that for every k there exists d = d(k) such that every graph of average degree at least d contains a subgraph of average degree at least k and girth at least six. This settles a special case of a conjecture of Thomassen.

Domination number of graph fractional powers

For any $k in mathbb{N}$, the $k$-subdivision of graph $G$ is a simple graph $G^{frac{1}{k}}$, which is constructed by replacing each edge of $G$ with a path of length $k$. In [Moharram N. Iradmusa, On colorings of graph fractional powers, Discrete Math., (310) 2010, No. 10-11, 1551-1556] the $m$th power of the $n$-subdivision of $G$ has been introduced as a fractional power of $G$, denoted by ...

The average degree in a vertex-magic graph

Graph Powers

The investigation of the asymptotic behaviour of various parameters of powers of a fixed graph leads to many fascinating problems, some of which are motivated by questions in information theory, communication complexity, geometry and Ramsey theory. In this survey we discuss these problems and describe the techniques used in their study which combine combinatorial, geometric, probabilistic and l...