On maximum graphs in Tutte polynomial posets
Nathan Kahl, Kristi Luttrell

TL;DR
This paper demonstrates that certain graphs maximizing all-terminal reliability also maximize a broad class of Tutte polynomial-derived parameters, unifying their optimality through Tutte polynomial posets.
Contribution
It establishes that UOR graphs are maximum for various Tutte polynomial-based parameters, extending their optimality beyond reliability to orientations, chromatic and flow polynomials, and generating functions.
Findings
UOR graphs are maximum for multiple Tutte polynomial parameters
Maximality is unified via Tutte polynomial posets
Results extend to orientations, chromatic and flow polynomials
Abstract
Boesch, Li, and Suffel were the first to identify the existence of uniformly optimally reliable graphs (UOR graphs), graphs which maximize all-terminal reliability over all graphs with vertices and edges. The all-terminal reliability of a graph, and more generally a graph's all-terminal reliability polynomial , may both be obtained via the Tutte polynomial of the graph . Here we show that the UOR graphs found earlier are in fact maximum graphs for the Tutte polynomial itself, in the sense that they are maximum not just for all-terminal reliability but for a vast array of other parameters and polynomials that may be obtained from as well. These parameters include, but are not limited to, enumerations of a wide variety of well-known orientations, partial orientations, and fourientations of ; the magnitudes of the coefficients of the chromatic…
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Taxonomy
TopicsAdvanced Combinatorial Mathematics · Graph Labeling and Dimension Problems · Graph theory and applications
