Exploring Zig-Zag Triangles: H-Coverings and Practical Applications

Graph theory is widely used to analyze the structure models in chemistry, biology, computer science, operations research and sociology. Molecular bonds, species movement between regions, development of computer algorithms, shortest spanning tree in a weighted graphs, aircraft scheduling and exploration of diffusion mechanism are some of these structure models. A computing system C can execute an algorithm A if A is graph-isomorphic to a subgraph of CG. This is equivalent of defining a graph isomorphism between A and subgraph of CG. This research article gives a small description of graph theory applications and one of graph model (zig-zag triangle) to define a suitable computer algorithm. Let Γ be a connected, simple graph with finite vertices v and edges e. A family {Γ1, Γ2, . . . , Γp} ⊂ Γ of subgraphs such that for all e ∈ E, e ∈ Γl, for some l, l = 1, 2, . . . , p is an edge-covering of Γ. If Γl ∼= Γ′, ∀l, then Γ has an Γ′-covering. Graph Γ with Γ′-covering is an (ad, d)-Γ′-antimagic if f : V ∪ E → {1, 2, . . . , |V | + |E|} a bijection exists and the sum over all vertex-weights and edge-weights of Γ′ form a set {ad, ad + d, . . . , ad + (p − 1)d}. The labeling ξ is super for ξ(VΓ) = {1, 2, 3, . . . , |VΓ|} and graph Γ is Γ′-supermagic for d = 0. This manuscript investigates super (ad, d)-Γ′-antimagic labelings of zig-zag triangles for differences d = 1, 2, . . . , 8.