A C-EXPONENTIAL MEAN LABELING OF CORONA, CARTESIAN AND TENSOR PRODUCT OF GRAPHS

A C-EXPONENTIAL MEAN LABELING OF CORONA, CARTESIAN AND TENSOR PRODUCT OF GRAPHS

The present work proposes a new C-exponential mean labeling (in short EML) of various product graphs. In this C- EML of the corona, tensor and cartesian product of paths and cycle are studied. Where the graphs are considered as simple, finite, and undirect graphs H(V, E), with s vertices and t edges. The main objective of the current study is to achieve integer values for labeling from the ceiling function, dispute of the decimal values of floor function which was obtained prior by others. To achieve this, the following functions are considered where the function 𝜙 is referred as a C-EML of a graph H with s vertices and t edges. If 𝜙 : V (H) → {1, 2, 3, . . . , t + 1} is one-one and the induced function 𝜙^∗ : E(H) → {2, 3, 4, . . . , t+1} defined as $${\phi}^*(xy)={\left\lceil\frac{1}{e}\;{\left(\begin{array}{c}\frac{{\phi}(y)^{{\phi}(y)}}{{\phi}(x)^{{\phi}(x)}}\end{array}\right)}\;^{\frac{1}{{\phi}(y)-{{\phi}(x)}}}\right\rceil},$$ for every bijective xy ∈ E(H), Where the number e is a mathematical constant approximately equal to 2.71828. In this paper, we have evaluated the C-exponential meanness of some product related graphs. We proved that the C- EML of the corona product of $P_s{\circ}{\bar{K}}_r$, s is even, $P_s{\circ}{\bar{K}}_2$, s is odd, C_r ∘ K_n, r ≥ 3 and n ≤ 2, C_s ∘ rK₁, s ≡ 0, 2 (mod 4) and C_s ∘ 2K₁, s ≡ 1, 3 (mod 4). Also proved that the C- EML of the cartesion product of C_r × P_s, r ≥ 3 and s ≥ 2, and book graph B_r. In addition, we have presented the C- EML of tensor products of C_s ⊗ P₂, s ≥ 4, is even.