The $\rho$-Capacity of a Graph

TL;DR

This paper introduces the $ ho$-capacity, a new graph capacity measure extending Shannon capacity, with bounds, properties under graph operations, and structural insights, motivated by zero-error broadcasting.

Contribution

It defines the $ ho$-capacity, derives bounds including a Lovász-type bound for regular graphs, and explores its behavior under graph operations and structure.

Findings

Derived upper and lower bounds on $ ho$-capacity.

Established a Lovász-type upper bound for regular graphs.

Analyzed the effect of graph operations on $ ho$-capacity.

Abstract

Motivated by the problem of zero-error broadcasting, we introduce a new notion of graph capacity, termed $\rho$-capacity, that generalizes the Shannon capacity of a graph. We derive upper and lower bounds on the $\rho$-capacity of arbitrary graphs, and provide a Lov\'asz-type upper bound for regular graphs. We study the behavior of the $\rho$-capacity under two graph operations: the strong product and the disjoint union. Finally, we investigate the connection between the structure of a graph and its $\rho$-capacity.