Generalizing K\"orner's graph entropy to graphons

arXiv (Cornell University), 2022

The first degree-based entropy of a graph is the Shannon entropy of its degree sequence normalized by the degree sum. Its correct interpretation as a measure of uniformity of the degree sequence requires the determination of its extremal values given natural constraints. In this paper, we prove that the graphs with given size that minimize the first degree-based entropy are the colex graphs.

2007 IEEE International Symposium on Information Theory, 2007

Let Aq(n, d) be the maximum order (maximum number of codewords) of a q-ary code of length n and Hamming distance at least d. And let A(n, d, w) that of a binary code of constant weight w. Building on results from algebraic graph theory and Erdős-ko-Rado like theorems in extremal combinatorics, we show how several known bounds on Aq(n, d) and A(n, d, w) can be easily obtained in a single framework. For instance, both the Hamming and Singleton bounds can derived as an application of a property relating the clique number and the independence number of vertex transitive graphs. Using the same techniques, we also derive some new bounds and present some additional applications.

Graphs and Combinatorics, 2022

The cochromatic number Z(G) of a graph G is the fewest number of colors needed to color the vertices of G so that each color class is a clique or an independent set. In a fractional cocoloring of G a non-negative weight is assigned to each clique and independent set so that for each vertex v, the sum of the weights of all cliques and independent sets containing v is at least one. The smallest total weight of such a fractional cocoloring of G is the fractional cochromatic number Z f (G). In this paper we prove results for the fractional cochromatic number Z f (G) that parallel results for Z(G) and the well studied fractional chromatic number χ f (G). For example Z f (G) = χ f (G) when G is triangle-free, except when the only nontrivial component of G is a star. More generally, if G contains no k-clique, then Z f (G) ≤ χ f (G) ≤ Z f (G) + R(k, k). Moreover, every graph G with χ f (G) = m contains a subgraph H with Z f (H) ≥ (1 4 − o(1)) m log 2 m. We also prove that the maximum value of Z f (G) over all graphs G of order n is Θ(n/ log n), and the maximum over all graphs embedded on an orientable surface of genus g is Θ(√ g/ log g).

Classical, Semi-classical and Quantum Noise, 2011

This paper is a tutorial on the application of graph theoretic techniques in classical coding theory. A fundamental problem in coding theory is to determine the maximum size of a code satisfying a given minimum Hamming distance. This problem is thought to be extremely hard and still not completely solved. In addition to a number of closed form expressions for special cases and some numerical results, several relevant bounds have been derived over the years.

joint-research.org

In this paper, we state information inequalities for nanostructures representing graphs by using some novel information functionals. We use a recently proposed approach to determine the structural information content of arbitrary undirected and connected graphs. In contrast to the information indices often used in chemical information theory, the entropy measure does not depend on the problem to determine vertex partitions of a graph under consideration. Finally, to define the entropy of a graph, we use certain information functionals. As the main result, we derive so-called implicit and explicit information inequalities for arbitrary undirected and connected graphs.

—Under study are the hereditary classes of edge-colored graphs. A class is called entropy minimal if it contains no proper hereditary subclasses with the same value of entropy (logarithmic density). For ordinary graphs it is known that, for all fixed a and b, the class consisting of all graphs whose vertex sets can be partitioned into a cliques and b independent sets is entropy minimal. We generalize this to colored graphs.

Journal of Combinatorial Theory, Series B, 1986

A polynomially computable upper bound for the weighted independence number of a graph is studied. This gives rise to a convex body containing the vertex packing polytope of the graph. This body is a polytope if and only if the graph is perfect. As an application of these notions, we show that the maximum weight independent set of an h-perfect graph can be found in polynomial time. ,. 1986 Academic Press, Inc. I. VERTEX PACKING AND ITS RELAXATIONS Throughout the paper we assume that all graphs we consider have no loops, no multiple edges, and are connected. For our purposes, these assumptions can be made without loss of generality. Let G be a graph and w: V(G)-+ IR + any weighting of its nodes. Let a(G; w) denote the maximum weight of an independent set. It is well known that to determine a(G; w) even in the special case when w == 1 is vf'Jl>-hard. A well-known approach to study a(G; w) is to introduce the vertex packing polytope VP(G), which is defined as the convex hull of incidence vectors • The first two authors were partially supported by the Hungarian Academy of Sciences and the Deutsche Forschungsgemeinschaft.

European Journal of Combinatorics, 2020

We study graphon counterparts of the chromatic and the clique number, the fractional chromatic number, the b-chromatic number, and the fractional clique number. We establish some basic properties of the independence set polytope in the graphon setting, and duality properties between the fractional chromatic number and the fractional clique number. We present a notion of perfect graphons and characterize them in terms of induced densities of odd cycles and its complements.

The Bulletin of Mathematical Biophysics, 1968

The connection between the adjacency matrix and the automorphisms of a digraph is used to develop a method for studying the automorphlsm group and, thus, the information content (Mowshowitz 1968a, b) of a digraph. An algorithm is given for constructing digraphs with zero information content, and the properties of such digraphs are examined. Moreover, an algorithm for computing the automorphism group of a digraph is presented and is used to find conditions which insure that two digraphs have the same information content. This algorithm is further used to detelmaine the information content of digraphs whose adjacency matrices have prescribed properties.

2004

We present two sequences of ensembles of non-systematic irregular repeat-accumulate codes which asymptotically (as their block length tends to infinity) achieve capacity on the binary erasure channel (BEC) with bounded complexity per information bit. This is in contrast to all previous constructions of capacity-achieving sequences of ensembles whose complexity grows at least like the log of the inverse of the gap (in rate) to capacity. The new bounded complexity result is achieved by puncturing bits, and allowing in this way a sufficient number of state nodes in the Tanner graph representing the codes. We also derive an information-theoretic lower bound on the decoding complexity of randomly punctured codes on graphs. The bound holds for every memoryless binary-input output-symmetric channel, and is refined for the BEC.