Spectral Energy of Semigraphs Induced by Binary Vertex Labeling: Bounds and Computations

Let S= (V (S) ) be a semigraph with vertex set V (S) and semiedge set E (S) and consider the set A=0, 1 A mapping ℓ: V (S) →A is called a binary vertex labeling of the semigraph S and ℓ (v) is referred to as the label of the vertex v under ℓ. In this paper, we introduce a new type of energy associated with binary labeled semigraphs, called the labeled semigraph energy, denoted by Eℓ (S). This energy depends on both the underlying semigraph structure and the chosen binary labeling. Upper and lower bounds for Eℓ (S) are established. Furthermore, the labeled energies of several well-known and commonly studied families of semigraphs are explicitly computed to illustrate the behavior and applicability of the proposed concept.