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The combinatorial problem in this paper is motivated by a variant of the famous traveling salesman problem where the salesman must return to the starting point after each delivery. The total length of a delivery route is related to a metric known as closeness centrality. The closeness centrality of a vertex v in a graph G was defined in 1950 by Bavelas to be CC(v)=|V(G)|−1SD(v), where SD(v) is the sum of the distances from v to each of the other vertices (which is one-half of the total distance in the delivery route). We provide a real-world example involving the Metro Atlanta Rapid Transit Authority rail network and identify stations whose SD values are nearly identical, meaning they have a similar proximity to other stations in the network. We then consider theoretical aspects involving asymmetric trees. For integer values of k, we considered the asymmetric tree with paths of lengths k,2k,…,nk that are incident to a center vertex. We investigated trees with different values of k, and for k=1 and k=2, we established necessary and sufficient conditions for the existence of two vertices with identical SD values, which has a surprising connection to the triangular numbers. Additionally, we investigated asymmetric trees with paths incident to two vertices and found a sufficient condition for vertices to have equal SD values. This leads to new combinatorial proofs of identities arising from Pascal’s triangle. 

For a given a graph G, the zero forcing number of G, Z(G), is the smallest cardinality of any set S of vertices on which repeated applications of the forcing rule results in all vertices being included in S. The forcing rule is as follows: if a vertex v is in S, and exactly one neighbor u of v is not in S, then u is added to S in the next iteration. The failed zero forcing number of a graph was introduced by Fetcie, Jacob, and Saavedra and was defined as the cardinality of the largest set of vertices which fails to force all vertices in the graph. In 2021, Gomez et al. proved that there were exactly 15 graphs with a failed zero forcing number of two, but their proof was complicated, requiring the analysis of many graph families. We present an “inverse” approach where we start with a set of vertices S and then see which graphs have S as a maximum-sized failed zero forcing set. This results in not only a shorter proof but characterizes which graphs have a particular failed zero forcing set. In our proof, we characterize the graphs which have a failed zero forcing set S where |S|=2, meaning considering all simple graphs of order two as induced subgraphs. This approach also has greater potential for characterizing graphs where F(G)>2 since many general arguments on the structure of graphs are developed in this type of analysis and are applicable for failed zero forcing sets of any size.