Kronecker Products, Polarity Quotients and Large Graph Constructions
Abstract
In this paper, we establish a structural compatibility between the Kronecker product of bipartite graphs that admit polarity and their polarity quotients, and provide a sharp upper bound on the diameter of these graphs. We use these properties to construct new families of graphs, all of which have orders that asymptotically approach the Moore bound. We show that the bipartite compatibility graphs have a natural polarity, and that the corresponding polarity quotients comprise a new family of graphs of any even diameter 2k with degree (q+1)k for any prime power q, much smaller than those of other known families of that diameter and field order q. These families generalize the Erdős-Rényi-Brown polarity graphs. Kronecker products of these polarity quotients give another new family of graphs of any even diameter for every pair of prime powers q,r. Certain generalized polygons are known to admit polarity with polarity quotients approaching the Moore bound asymptotically. Kronecker products of these polarity quotients give another new family of graphs of diameters $2$, $3$ and $5$, including a diameter-$3$ degree-$18$ graph of order larger than any previously known for that degree and diameter.
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BibTeX
@misc{isham2026kronecker,
title = {{Kronecker Products, Polarity Quotients and Large Graph Constructions}},
author = {Kelly Isham and Kartik Lakhotia and Laura Monroe and Fabrizio Petrini},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2608.27253},
eprint = {2608.27253},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.27253},
}
RIS
TY - UNPB
TI - Kronecker Products, Polarity Quotients and Large Graph Constructions
AU - Kelly Isham
AU - Kartik Lakhotia
AU - Laura Monroe
AU - Fabrizio Petrini
PY - 2026
DA - 2026/10/22
PB - arXiv
DO - 10.48550/arxiv.2608.27253
UR - https://arxiv.org/abs/2608.27253
AB - In this paper, we establish a structural compatibility between the Kronecker product of bipartite graphs that admit polarity and their polarity quotients, and provide a sharp upper bound on the diameter of these graphs. We use these properties to construct new families of graphs, all of which have orders that asymptotically approach the Moore bound. We show that the bipartite compatibility graphs have a natural polarity, and that the corresponding polarity quotients comprise a new family of graphs of any even diameter 2k with degree (q+1)^k for any prime power q, much smaller than those of other known families of that diameter and field order q. These families generalize the Erdős-Rényi-Brown polarity graphs. Kronecker products of these polarity quotients give another new family of graphs of any even diameter for every pair of prime powers q,r. Certain generalized polygons are known to admit polarity with polarity quotients approaching the Moore bound asymptotically. Kronecker products of these polarity quotients give another new family of graphs of diameters $2$, $3$ and $5$, including a diameter-$3$ degree-$18$ graph of order larger than any previously known for that degree and diameter.
ER -