# The energy of random signed graph

## TL;DR

This paper investigates the energy of random signed graphs, providing exact estimates for almost all such graphs and bounds for their multipartite variants, advancing understanding of spectral properties in signed network models.

## Contribution

It offers the first exact energy estimates for almost all random signed graphs and establishes bounds for their multipartite counterparts.

## Key findings

- Exact energy estimates for almost all random signed graphs.
- Lower and upper bounds for the energy of random multipartite signed graphs.
- Enhanced understanding of spectral properties in signed graph models.

## Abstract

A signed graph $\Gamma(G)$ is a graph with a sign attached to each of its edges, where $G$ is the underlying graph of $\Gamma(G)$. The energy of a signed graph $\Gamma(G)$ is the sum of the absolute values of the eigenvalues of the adjacency matrix $A(\Gamma(G))$ of $\Gamma(G)$. The random signed graph model $\mathcal{G}_n(p, q)$ is defined as follows: Let $p, q \ge 0$ be fixed, $0 \le p+q \le 1$. Given a set of $n$ vertices, between each pair of distinct vertices there is either a positive edge with probability $p$ or a negative edge with probability $q$, or else there is no edge with probability $1-(p+ q)$. The edges between different pairs of vertices are chosen independently. In this paper, we obtain an exact estimate of energy for almost all signed graphs. Furthermore, we establish lower and upper bounds to the energy of random multipartite signed graphs.