Related papers: Entropy-Based Proofs of Combinatorial Results on B…
We estimate Ramsey numbers for bipartite graphs with small bandwidth and bounded maximum degree. In particular we determine asymptotically the two and three color Ramsey numbers for grid graphs. More generally, we determine asymptotically…
An edge-coloring of a graph $G$ with colors $1,\ldots,t$ is an \emph{interval $t$-coloring} if all colors are used, and the colors of edges incident to each vertex of $G$ are distinct and form an integer interval. It is well-known that…
We prove that if $G$ is an $n$-vertex graph whose edges are coloured with red and blue, then the number of colour-alternating walks of length $2k+1$ with $k+1$ red edges and $k$ blue edges is at most $k^k(k+1)^{k+1}(2k+1)^{-2k-1}n^{2k+2}$.…
We initiate the study of property testing in arbitrary planar graphs. We prove that bipartiteness can be tested in constant time, improving on the previous bound of $\tilde{O}(\sqrt{n})$ for graphs on $n$ vertices. The constant-time…
A bipartite covering of a (multi)graph $G$ is a collection of bipartite graphs, so that each edge of $G$ belongs to at least one of them. The capacity of the covering is the sum of the numbers of vertices of these bipartite graphs. In this…
For the set of graphs with a given degree sequence, consisting of any number of $2's$ and $1's$, and its subset of bipartite graphs, we characterize the optimal graphs who maximize and minimize the number of $m$-matchings. We find the…
We obtain several new upper bounds of the odd graceful chromatic number of a graph $G$, which must be bipartite. Some of our bounds depend only on the number of the vertices of $G$ or the chromatic number of some graphs related to the…
This article proposes a method to quantify the structure of a bipartite graph using a network entropy per link. The network entropy of a bipartite graph with random links is calculated both numerically and theoretically. As an application…
The entropy of a graph is a functional depending both on the graph itself and on a probability distribution on its vertex set. This graph functional originated from the problem of source coding in information theory and was introduced by J.…
Classical spectral graph theory characterizes graphs with logarithmic mixing time. In this work, we present a combinatorial characterization of graphs with constant mixing time. The combinatorial characterization is based on the small-set…
Using the theory of combinatorial species, we compute the cycle index for bipartite graphs, which we use to count unlabeled bipartite graphs and bipartite blocks.
Topological drawings are natural representations of graphs in the plane, where vertices are represented by points, and edges by curves connecting the points. Topological drawings of complete graphs and of complete bipartite graphs have been…
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…
This paper proves limit theorems for the number of monochromatic edges in uniform random colorings of general random graphs. These can be seen as generalizations of the birthday problem (what is the chance that there are two friends with…
We prove lower bounds for the entanglement of formation and the squashed entanglement for any a bipartite density matrix in terms of the conditional entropy of the bipartite state with respect to either of its partial traces, and prove that…
Andrews and El Bachraoui recently studied various two-colored integer partitions, including those related to two-colored partitions into distinct parts with constraints and overpartitions. Their work raised questions about the existence of…
In this paper, we investigate the structural properties of trees and bipartite graphs through the lens of topological indices and combinatorial graph theory. We focus on the First and Second Hyper-Zagreb indices, $HM_1(G)$ and $HM_2(G)$,…
We use an entropy based method to study two graph maximization problems. We upper bound the number of matchings of fixed size $\ell$ in a $d$-regular graph on $N$ vertices. For $\frac{2\ell}{N}$ bounded away from 0 and 1, the logarithm of…
Given two orthornormal bases A and B, the basic form of the entropic uncertainty principle is stated in terms of the sum of the Shannon entropies of the probabilities of measuring A and B onto a given quantum state. State independent lower…
Inspired by notorious combinatorial optimization problems on graphs, in this paper we consider a series of related problems defined using a metric space and topology determined by a graph. Particularly, we present the Independent Set,…