SEMT Labelings and Deficiencies of Forests with Two Components (I)

Abstract

A set of nodes called vertices V accompanied with the lines that bridge these nodes called edges E compose an explicit figure termed as a graph G(V, E). |V (G)| = ν and |E(G)| = ε specify its order and size respectively. A (ν, ε)-graph G determines an edge-magic total (EMT) labeling when Γ : V (G) ∪ E(G) → {1, ν + ε} is bijective so as the weights at every edge are the same constant (say) c i.e., for x, y ∈ V (G); Γ(x) + Γ(xy) + Γ(y) = c, independent of the choice of any xy ∈ E(G), such a number is interpreted as a magic constant. If all vertices gain the smallest of the labels then an EMT labeling is called a super edge-magic total (SEMT) labeling. If a graph G allows at least one SEMT labeling then the smallest of the magic constants for all possible distinct SEMT labelings of G describes super edge-magic total (SEMT) strength, sm(G), of G. For any graph G, SEMT deficiency is the least number of isolated vertices which when uniting with G yields a SEMT graph. In this paper, we will find SEMT labeling and deficiency of forests consisting of two components, where one of the components for each forest is generalized comb Cbτ (`, `, . . . , ` | {z } τ−times ) and other component is a star, bistar, comb or path respectively, moreover, we will investigate SEMT strength of aforesaid generalized comb.