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

Let p be an odd prime and let Fi be the it (a, b)-Fibonacci number with initial values F0 = a and F1 = b. For a simple connected graph G = (V, E), define a bijective function f : V (G) → {0, 1, . . . , |V | − 1}. If the induced function f∗ p: E(G) → {0, 1}, defined by f∗p(uv) = 1+([Ff(u)+Ff(v)]/p)2 whenever Ff(u) +Ff(v) ̸≡ 0(mod p) and f∗p(uv) = 0 whenever Ff(u) + Ff(v) ≡ 0(mod p), satisfies the condition |ef∗p(0)−ef∗p(1)| ≤ 1 where ef∗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.