Unveiling New Cube Sum Graphs

Let G = (V, E) be a (p, q) graph, where p is the order and q is the size of G. An injective function f : V (G) −→ {1, 2, . . ., p is said to be a cube sum labeling if the induced each labeling function f* (uv) = f (u)³ + f (v)³ is injective. A graph admitting a cube sum labeling is called a cube sum graph. In this study, an investigation is conducted in to that the ladder graph L_n, the tadpole graph T (m, n), the cycle with triangle C_n(1, 1, n − 2), the k-polygonal snake graph S_n(C_k) for odd k and the one point union C_k⁽ⁿ⁾ as cube sum graph. These results involve determining whether the Diophantine equation a³ + b³ = c³ + d³ has an integer solution or not.