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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.

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Navigating recovery while postpartum: examining the relationship between maternal functioning and substance use disorders

Our current understanding of postpartum people’s functioning while managing substance use disorder (SUD) recovery is limited. This study explored whether having a SUD impacts maternal functioning and examined the reliability of the Barkin Index of Maternal Functioning (BIMF) in postpartum people with SUDs. We conducted a prospective, cross-sectional survey (n = 141) by administering the BIMF and a demographic questionnaire with a sample of postpartum people with SUD (n = 48) and a comparative sample of postpartum people without SUD (n = 93). The internal reliability (Cronbach’s alpha = .86) was consistent with previous studies. SUD diagnosis was significantly associated with lower functioning, with individuals in the SUD group having an 8-point lower BIMF score compared to persons without SUDs, although both groups reported moderately high BIMF scores (>80). While not statistically significant, participants with SUDs had reduced functioning scores over time. Considering the increasing rate of overdose deaths in postpartum persons, further examination of functioning in postpartum persons with SUDs is warranted. Integration of the BIMF into SUD screening and treatment can provide a holistic view of postpartum experiences in this high-risk population and serve as a unique tool to understand the needs of new mothers navigating SUD recovery.

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