For a simple connected graph G = (V, E), a bijective function f: V (G) → 1, 2,. . . , |V| is said to be a Legendre cordial labeling modulo p, where p is an odd prime, if the induced function fp ∗: E (G) → 0, 1 defined by fp ∗ (uv) = 0 if (ωuv/p) = −1 or ωuv = 0, and fp ∗ (uv) = 1 if (ωuv/p) = 1, where f (u) + f (v) ≡ ωuv (mod p), 0 ≤ ωuv < p, satisfies the condition efp * (0) − efp ∗ (1) ≤ 1, where efp ∗ (i) is the number of edges with label i (i = 0, 1). In this paper, we determined the values of n, m, and k so that the following special graphs admit a Legendre cordial labeling modulo p for an odd prime p: path graph Pn, cycle graph Cn, star graph Sn, tadpole graph Tn, m, kayak paddle graph KPn, m, k, bistar graph Bn, m, and fan graph Fn.
Andoyo et al. (Mon,) studied this question.