(a,b)-Fibonacci-Legendre Cordial Graphs and k-Pisano-Legendre Primes

TL;DR

This paper introduces a new graph labeling concept based on $(a,b)$-Fibonacci numbers and Legendre primes, exploring its properties on various graph classes and operations.

Contribution

It defines $(a,b)$-Fibonacci-Legendre cordial labeling, analyzes its application to common graphs, and investigates properties and distribution of related $k$-Pisano-Legendre primes.

Findings

Established conditions for $(a,b)$-Fibonacci-Legendre cordial labeling on several graph types.

Explored the relationship between these labelings and $k$-Pisano-Legendre primes.

Presented conjectures on the density and growth of $k$-Pisano-Legendre primes.

Abstract

Let $p$ be an odd prime and let $F_i$ be the $i$th $(a,b)$-Fibonacci number with initial values $F_0=a$ and $F_1=b$. For a simple connected graph $G=(V,E)$, define a bijective function $f:V(G)\to \{0,1,\ldots,|V|-1\}$. If the induced function $f_p^*:E(G)\to \{0,1\}$, defined by $f_p^*(uv)=\frac{1+([F_{f(u)}+F_{f(v)}]/p)}{2}$ whenever $F_{f(u)}+F_{f(v)}\not\equiv 0\pmod{p}$ and $f_p^*(uv)=0$ whenever $F_{f(u)}+F_{f(v)}\equiv 0\pmod{p}$, satisfies the condition $|e_{f_p^*}(0)-e_{f_p^*}(1)|\leq 1$ where $e_{f_p^*}(i)$ is the number of edges labeled $i$ ($i=0,1$), then $f$ is called $(a,b)$-Fibonacci-Legendre cordial labeling modulo $p$. In this paper, the $(a,b)$-Fibonacci-Legendre cordial labeling of path graphs, star graphs, wheel graphs, and graphs under the operations join, corona, lexicographic product, cartesian product, tensor product, and strong product is explored in relation to $k$-Pisano-Legendre primes relative to $(a,b)$. We also present some properties of $k$-Pisano-Legendre primes relative to $(a,b)$ and numerical observations on its distribution, leading to several conjectures concerning their density and growth behavior.