Some minimum broadcast graphs

Girth, minimum degree, independence, and broadcast independence

An independent broadcast on a connected graph $G$is a function $f:V(G)to mathbb{N}_0$such that, for every vertex $x$ of $G$, the value $f(x)$ is at most the eccentricity of $x$ in $G$,and $f(x)>0$ implies that $f(y)=0$ for every vertex $y$ of $G$ within distance at most $f(x)$ from $x$.The broadcast independence number $alpha_b(G)$ of $G$is the largest weight $sumlimits_{xin V(G)}f(x)$of an ind...

Minimum-energy Broadcast in Simple Graphs with Limited Node Power

Minimum-energy Broadcast in Simple Graphs with Limited Node Power

The minimum-energy broadcasting problem in wireless networks consists of finding a transmission radius vector for all stations in such a way that the total transmission power of the whole network is least possible. The minimum-energy broadcast problem may by modeled by an edge weighted complete graph in which each vertex in the graph represents a station and the weight of the edge is distance b...

Some special minimum k-geodetically connected graphs

A connected graph G is k-geodetically connected (k-GC) if the removal of less than k vertices does not affect the distances (lengths of the shortest paths) between any pair of the remaining vertices. As such graphs have important applications in robust system designs, we are interested in theminimumnumber of edges required tomake a k-GC graph of order n, and characterizing those minimum k-GC gr...

Minimum Tenacity of Toroidal graphs

The tenacity of a graph G, T(G), is dened by T(G) = min{[|S|+τ(G-S)]/[ω(G-S)]}, where the minimum is taken over all vertex cutsets S of G. We dene τ(G - S) to be the number of the vertices in the largest component of the graph G - S, and ω(G - S) be the number of components of G - S.In this paper a lower bound for the tenacity T(G) of a graph with genus γ(G) is obtained using the graph's connec...