A Sharp Upper Bound for the Boundary Independence Broadcast Number of a Tree

TL;DR

This paper establishes a precise upper limit for the boundary independence broadcast number in trees, linking it to the tree's size and specific branch vertex characteristics.

Contribution

It introduces a new sharp upper bound for the boundary independence broadcast number in trees, based on their order and branch vertex properties.

Findings

Derived a sharp upper bound for {}n(T) in trees.

Connected the bound to tree order and branch vertex count.

Provides theoretical insight into broadcast independence in trees.

Abstract

A broadcast on a nontrivial connected graph G with vertex set V is a function f from V to {0,1,...,diam(G)} such that f(v) is at most the eccentricity of v for all vertices v. The weight of f is the sum of the function values taken over V. A vertex u hears f from v if f(v) is positive and d(u,v) is at most f(v). A broadcast f is boundary independent if, for any vertex w that hears f from vertices v_{1},...,v_{k}, where k is at least 2, d(w,v_{i}) equals f(v_{i}) for each i. The maximum weight of a boundary independent broadcast on G is denoted by {\alpha}_{bn}(G). We prove a sharp upper bound on {\alpha}_{bn}(T) for a tree T in terms of its order and number of branch vertices of a certain type.