Weighted Padovan graphs

TL;DR

This paper introduces weighted Padovan graphs based on Padovan words, analyzes their structural properties, and proves they are median graphs, expanding understanding of their combinatorial and algebraic features.

Contribution

The paper defines weighted Padovan graphs, explores their properties, and proves they are median graphs, providing new insights into their structure and symmetries.

Findings

Determined order, size, degree, diameter, cube polynomial, automorphism group.

Presented two isomorphic families of weighted Padovan graphs.

Proved that weighted Padovan graphs are median graphs.

Abstract

Weighted Padovan graphs $\Phi^{n}_{k}$, $n \geq 1$, $\lfloor \frac{n}{2} \rfloor \leq k \leq \lfloor \frac{2n-2}{3} \rfloor$, are introduced as the graphs whose vertices are all Padovan words of length $n$ with $k$ $1$s, two vertices being adjacent if one can be obtained from the other by replacing exactly one $01$ with a $10$. By definition, $\sum_k |V(\Phi^{n}_{k})|=P_{n+2}$, where $P_n$ is the $n$th Padovan number. Two families of graphs isomorphic to weighted Padovan graphs are presented. The order, the size, the degree, the diameter, the cube polynomial, and the automorphism group of weighted Padovan graphs are determined. It is also proved that they are median graphs.