From Independent Sets and Vertex Colorings to Isotropic Spaces and Isotropic Decompositions: Another Bridge between Graphs and Alternating Matrix Spaces

Abstract

In the 1970's, Lovász built a bridge between graphs and alternating matrix spaces, in the context of perfect matchings (FCT 1979). A similar connection between bipartite graphs and matrix spaces plays a key role in the recent resolutions of the non-commutative rank problem (Garg-Gurvits-Oliveira-Wigderson, FOCS 2016; Ivanyos-Qiao-Subrahmanyam, ITCS 2017). In this paper, we lay the foundation for another bridge between graphs and alternating matrix spaces, in the context of independent sets and vertex colorings. The corresponding structures in alternating matrix spaces are isotropic spaces and isotropic decompositions, both useful structures in group theory and manifold theory.We first show that the maximum independent set problem and the vertex c-coloring problem reduce to the maximum isotropic space problem and the isotropic c-decomposition problem, respectively. Next, we show that several topics and results about independent sets and vertex colorings have natural correspondences for isotropic spaces and decompositions. These include algorithmic problems, such as the maximum independent set problem for bipartite graphs, and exact exponential-time algorithms for the chromatic number, as well as mathematical questions, such as the number of maximal independent sets, and the relation between the maximum degree and the chromatic number. These connections lead to new interactions between graph theory and algebra. Some results have concrete applications to group theory and manifold theory, and we initiate a variant of these structures in the context of quantum information theory. Finally, we propose several open questions for further exploration.

Topological Evaluation of Certain Computer Networks by Contraharmonic-Quadratic Indices

“…The purpose of this study is to evaluate the topology of bridge networks before developing and using them in any field [3]. Bridge graphs [4] are introduced by Mansour et al [5] which is a mix of networks bridged together in a single network. These are also called certain computer networks.…”

“…It is done because of the automorphism property of the graph. In the fields of computer sciences, chemistry, mathematics and robotics there is a ton of utilizations of the graph hypothesis [4].…”

“…QSAR and QSPR are giving the establishment to these models. The last comment is that the use of the estimation in the network plane works with a quantitative assessment of different geography shielding planning calculations [4].…”

K-Banhatti Invariants Empowered Topological Investigation of Bridge Networks

“…These are used in the construction of memory interconnection, microchips, integrated circuits, power generation interconnection, robotics interconnection and chemical structures. A bridge graph is a graph acquired from number of graphs G1, G2, G3, Gm by partner the vertices vi and vi + 1 by an edge∀, I = 1, 2,., m − 1 [1]. V. R. Kulli in 2016, defined the possibility of various types of K-Banhatti invariants.…”