Abstract
The Shields‐Harary numbers are a class of graph parameters that measure a certain kind of robustness of a graph, thought of as a network of fortified reservoirs, with reference to a given cost function. We prove a result about the Shields‐Harary numbers with respect to concave continuous cost functions which will simplify the calculation of these numbers for certain classes of graphs, including graphs formed by two intersecting cliques, and complete multipartite graphs.
Highlights
Suppose we have finite simple graph G and a “weighting” function g : V (G) → [0, ∞), which together constitute a weighted network
Think of the weights assigned to each vertex of G by g as representing some amount of harmful “stuff” stored there
Some enemy of this weighted network might wish to dismantle it by knocking out vertices until the sum of weights on each remaining connected component is no greater than some threshold, say 1. (We will call a set of vertices which, after being knocked out, satisfies this requirement a g-dismantling set.) The enemy does not get to knock out vertices for free
Summary
Suppose we have finite simple graph G and a “weighting” function g : V (G) → [0, ∞), which together constitute a weighted network. Think of the weights assigned to each vertex of G by g as representing some amount of harmful “stuff” stored there Some enemy of this weighted network might wish to dismantle it by knocking out vertices until the sum of weights on each remaining connected component is no greater than some threshold, say 1. Johnson [3] presented everything known at the time about the SH parameters with that particular cost function. The following conjecture is posed by Harary and Johnson: if f is continuous and G is vertex-transitive, there is a constant optimal weighting of V (G). We end the paper with examples of how we apply these results to obtain the Shields-Harary numbers of some graphs with particular concave cost functions
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