# Bounds for the completely positive rank of a symmetric matrix over a tropical semiring

## TL;DR

This paper establishes an upper bound for the completely positive rank of symmetric matrices over tropical semirings based on vertex clique covers, and characterizes graphs with minimal CP-rank as having diameter two.

## Contribution

It introduces a bound for CP-rank tied to graph clique covers and characterizes graphs with minimal CP-rank as having diameter two.

## Key findings

- Upper bound for CP-rank based on vertex clique cover
- Graphs with minimal CP-rank have diameter two
- Characterization of graph patterns with lowest CP-rank

## Abstract

In this paper, we find an upper bound for the CP-rank of a matrix over a tropical semiring, according to the vertex clique cover of the graph prescribed by the pattern of the matrix. We study the graphs that beget the patterns of matrices with the lowest possible CP-ranks and prove that any such graph must have its diameter equal to 2.