On the nullspace of split graphs

TL;DR

This paper characterizes the nullspace of split graphs' adjacency matrices, introducing the clique-kernel and providing formulas linking nullity to submatrix properties, advancing algebraic understanding of graph singularity.

Contribution

It introduces the clique-kernel concept, derives a nullity formula, and analyzes nullspace behavior under graph composition, offering a unified algebraic framework for split graphs.

Findings

Nullity of split graphs expressed as null(R) + dimension of clique-kernel

Clique-kernel dimension is at most one, simplifying nullspace analysis

Closed formula for the determinant of split graphs

Abstract

We study the nullspace of the adjacency matrix of split graphs, whose vertex set can be partitioned into a clique and an independent set. We introduce the clique-kernel, a subspace that decides whether clique vertices lie in the support of a kernel eigenvector, and we prove that its dimension is at most one. This yields the formula $null(Sp) = null(R) + \dim(\mathrm{Cker}(Sp))$, which fully describes the nullity of a split graph in terms of the biadjacency submatrix $R$. We also analyze unbalanced split graphs through the concept of swing vertices and characterize the structure of their kernel supports. Furthermore, we study the behavior of the nullspace under Tyshkevich composition and derive a closed formula for the determinant. These results provide a unified algebraic framework for understanding when a split graph is singular and how its combinatorial structure determines its nullspace.