Problem
Research questions and friction points this paper is trying to address.
Kronecker product
polarity quotient
large graph construction
diameter upper bound
generalized polygons
Kronecker Products, Polarity Quotients and Large Graph Constructions
🤖 AI Summary
本文通过研究Kronecker积与极性商图的结构相容性，提供了一种构建具有大阶数且直径分别为2、3和5的新图族的方法。
📝 Abstract
In this paper, we establish a structural compatibility between the Kronecker product of bipartite graphs that admit polarity and their polarity quotient, and provide a sharp upper bound on the diameter of these graphs.
For certain factor graphs, the diameter of the Kronecker product meets the upper bound on diameter, among them the generalized polygons. Generalized polygons with their polarity quotients have been notably used in the past to construct very large graphs. We apply the structural theorems in the paper to generalized polygons $\mathbb{G}_n(q,q)$ used as factor graphs, and build three new families of graphs of large order covering an infinite but sparse set of degrees, one of diameter $2$, one of diameter $3$ and one of diameter $5$.
These asymptotically approach a theoretical upper bound on graph size as orders $q$ and $r$ of the generalized polygon factors increase. As an example, we develop one such family, derived from generalized quadrangles, and construct new diameter-$3$ graphs of low degree that are larger than any previously known at their degrees.
Innovation
Methods, ideas, or system contributions that make the work stand out.
Kronecker product
polarity quotient
generalized polygons
diameter bound
large graph construction
🔎 Similar Papers
No similar papers found.
