(a,b)-Fibonacci-Legendre Cordial Graphs and k-Pisano-Legendre Primes
J. D. Andoyo
TL;DR
Analyzes a fusion of Fibonacci-type sequences with Legendre-symbol based cordial labeling modulo an odd prime p and defines k-Pisano-Legendre primes relative to (a,b). Develops constructive labelings for path, star, and wheel graphs and for standard graph operations under the modular framework, including a 0-PL prime condition and parity considerations. Introduces the (a,b)-Pisano period pi_p(a,b), Legendre partitions Lambda_j^p(a,b), and the minimal-prime function zeta_(a,b)(k), supported by numerical data and conjectures on density and growth. Connects combinatorial labeling with number-theoretic periodicity, outlining new conjectures and directions for analytic and computational investigation.
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.