# Maximum nullity of Cayley graph

## TL;DR

This paper establishes a lower bound for the maximum nullity of Cayley graphs using group representation theory, specifically focusing on cyclic groups and their products, and leverages Babai's spectrum result.

## Contribution

It introduces a new lower bound for Cayley graph nullity based on irreducible characters, extending previous spectral analysis methods.

## Key findings

- Provides a lower bound for maximum nullity of Cayley graphs.
- Uses representation theory and characters of groups.
- Determines the spectrum of Cayley graphs for specific group structures.

## Abstract

One of the most interesting problems on maximum nullity (minimum rank) is to characterize $M(\mathcal{G})$ ($mr(\mathcal{G})$) for a graph $\mathcal{G}$. In this regard, many researchers have been trying to find an upper or lower bound for the maximum nullity. For more results on this topic, see \cite{4}, \cite{2}, \cite{10} and \cite{1}. In this paper, by using a result of Babai \cite{Babai}, which presents the spectrum of a Cayley graph in terms of irreducible characters of the underlying group, and using representation and character of groups, we give a lower bound for the maximum nullity of Cayley graph, $X_S(G)$, where $G=\langle a\rangle$ is a cyclic group, or $G=G_1\times \cdots\times G_t$ such that $G_1=\langle a\rangle$ is a cyclic group and $G_i$ is an arbitrary finite group, for some $2\leq i\leq t$, with determine the spectrum of Cayley graphs.