Abstract

We consider a variant of the well-known Traveling Salesman Problem where the salesman must return to the starting point after each delivery. Ramanathan et al. asked whether different starting locations in a network yield the same total travel distance. The transmission of a vertex $u$ is defined to be $T(u)=\sum\limits_{v\in V(G)}d(u,v)$ where $d(u,v)$ is the number of edges in a shortest path between $u$ and $v$.  For specialized families of graphs, we investigate necessary and sufficient conditions for a graph to have distinct transmissions.  Graph asymmetry is a necessary condition for unique transmissions. However, it is not sufficient, as asymmetric graphs may still contain vertices with identical transmission values. Motivated by past constructions, we introduce a family of asymmetric graphs that possess unique transmissions and analyze how this property behaves under structural modifications. We call this family "noodle arm graphs." We further examine connections between unique transmissions and the Wiener Index, which can be expressed as half the sum of all vertex transmissions. Finally, we extend our constructions to a generalization of cycle graphs with attached pendant paths, within which the noodle arm graphs form a subfamily, providing a broader framework for analyzing transmission uniqueness in unicyclic graphs.

Publication Date

4-27-2026

Document Type

Thesis

Student Type

Graduate

Degree Name

Applied and Computational Mathematics (MS)

Department, Program, or Center

Mathematics and Statistics, School of

College

College of Science

Advisor

Darren Narayan

Advisor/Committee Member

Jobby Jacob

Advisor/Committee Member

Brendan Rooney

Campus

RIT – Main Campus

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