chromatic numberindependence numberIt was proved (A. Kotlov and L. Lovász, The rank and size of graphs, J. Graph Theory 23(1996), 185–189) that the number of vertices in a twin‐free graph is $O((\\\sqrt{2})^r)$ where r is the rank of the adjacency matrix. This bound was...
Some Relations between Rank, Chromatic Number and Energy of Graphs The energy of a graph $G$, denoted by $E(G)$, is defined as the sum of the\nabsolute values of all eigenvalues of $G$. Let $G$ be a graph of order $n$ and\n${m rank}(G)$ be the rank of the adjacency mat...
They constructed a graph with chromatic number 32 and with an adjacency matrix of rank 29. It was proved by Kotlov and Lovász, [6], that the number of vertices in a twin free graph (a graph with no two vertices with the same set of neighbors) is O((2)rk(G)), where rk(G) is...
The chromatic index of a graph whose core hascycle ranktwo is discussed. 本文讨论核的圈秩为2的简单图的边着色分类问题。 2) Cyclomatic number 圈秩 1. For any integer n,an infinite family of planar and bipartite chemical graphs with cyclomatic number two are constructed such that their line gra...
et al. The physics of higher-order interactions in complex systems. Nat. Phys. 17, 1093 (2021). Article CAS Google Scholar Wilf, H. S. The eigenvalues of a graph and its chromatic number. J. Lond. Math. Soc. 1, 330 (1967). Article MathSciNet Google Scholar Donath, W. E. &...
Computing the smallest number q such that the vertices of a given graph can be properly q -colored, known as the chromatic number , is one of the oldest and most fundamental problems in combinatorial optimization. The q -Coloring problem has been studied intensively using the framework of param...
Equivalently, it says that the chromatic number of a graph is bounded quasi-polynomially in the rank of its adjacency matrix. This old conjecture is well known among computer scientists and mathematicians, but despite extensive work it is still wide open. We survey results relating to the log-...
We show that a connected graph with real flow number p / q +1, where p and q are two relatively prime numbers must have cycle rank at least p + q - 1. A special case of this result yields that the real flow number of a 2-connected cubic graph with chromatic index 4 and order ...
In the end, the Schein rank of a fuzzy matrix has been considered by the view of general decomposition. For a given fuzzy matrix, it is proved that its Schein rank is equal to the chromatic number of a simple graph generated by it. Thus, calculating the Schein rank of a fuzzy matrix ...
The mammary glands of age-matched females with concomitant ablation of Rank appeared largely nor- mal, displaying a significantly lower number of MINs and no detectable carcinomas (Figure 1A-1C and Sup- plementary information, Figure S3). Quantification of branching points in whole mount staining...