Annals of Communications in Mathematics

ABSTRACT. Let \( n \ge 3 \) be an integer with primitive root \( \varpi \). For a simple connected graph \( G \) of order \( n \), a bijective function \( f : V(G) \to \{1,2,\ldots,n\} \) is called a logarithmic cordial labeling to the base \( \varpi \) modulo \( n \) if the induced function \( f_{\varpi,n}^{*} : E(G) \to \{0,1\} \) is defined by\[f_{\varpi,n}^{*}(ab)=\begin{cases}0, \text{ if } \mathrm{ind}_{\varpi,n}(f(a)+f(b)) \equiv 0 \pmod 2 \text{ or } \gcd(f(a)+f(b),n)\neq 1, \\1, \text{ if } \mathrm{ind}_{\varpi,n}(f(a)+f(b)) \equiv 1 \pmod 2,\end{cases}\]and satisfies the condition \( |e_{f_{\varpi,n}}(0) - e_{f_{\varpi,n}}(1)| \le 1 \), where \( e_{f_{\varpi,n}}(i) \) is the number of edges with label \( i \ (i=0,1) \).

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 \( f : V(G) \rightarrow \{0,1,\ldots,|V|-1\} \). If the induced function \( f_p^* : E(G) \rightarrow \{0,1\} \), defined by \( f_p^*(uv) = \frac{1+((F_{f(u)}+F_{f(v)})/p)}{2} \) whenever \( F_{f(u)} + F_{f(v)} \not\equiv 0 \; (\text{mod } p) \) and \( f_p^*(uv) = 0 \) whenever \( F_{f(u)} + F_{f(v)} \equiv 0 \; (\text{mod } p) \), satisfies the condition \( |e_f^*(0) - e_f^*(1)| \leq 1 \) where \( e_f^*(i) \) is the number of edges labeled \( i \) (\( i = 0,1 \)), then \( f \) is called a \( (a,b) \)-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.