A Spinorial Formulation of the Maximum Clique Problem of a Graph

TL;DR

This paper introduces a novel complex space and spinorial framework for the maximum clique problem, leveraging Clifford algebra and pure spinor geometry to potentially advance solution methods.

Contribution

It formulates the maximum clique problem using spinorial language and Clifford algebra, providing a new geometric perspective for this combinatorial challenge.

Findings

Graph adjacency matrices can be expressed via complex null vectors.

Maximum clique problem is transformed into a geometric problem in spinor space.

New formulation may enable novel solution approaches based on pure spinor geometry.

Abstract

We present a new formulation of the maximum clique problem of a graph in complex space. We start observing that the adjacency matrix A of a graph can always be written in the form A = B B where B is a complex, symmetric matrix formed by vectors of zero length (null vectors) and the maximum clique problem can be transformed in a geometrical problem for these vectors. This problem, in turn, is translated in spinorial language and we show that each graph uniquely identifies a set of pure spinors, that is vectors of the endomorphism space of Clifford algebras, and the maximum clique problem is formalized in this setting so that, this much studied problem, may take advantage from recent progresses of pure spinor geometry.