Weighted Padovan graphs

Weighted Padovan graphs ${\mathrm{\Phi }}_k^n$, $n\ge 1$, $\lfloor \frac{n}{2}\rfloor \le k\le \lfloor \frac{2n-2}{3}\rfloor$, 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, ${\sum }_k\mathrm{\mid }V({\mathrm{\Phi }}_k^n)\mathrm{\mid }=P_{n+2}$, 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.