On H-antimagic coverings for m-shadow and closed m-shadow of connected graphs

Abstract

An (𝑎, 𝑑)-𝐻-antimagic total labeling of a simple graph 𝐺 admitting an 𝐻-covering is a bijection 𝜑 ∶ 𝑉 (𝐺) ∪ 𝐸(𝐺) → {1, 2, … , |𝑉 (𝐺)| + |𝐸(𝐺)|} such that for all subgraphs 𝐻 ′ of 𝐺 isomorphic to 𝐻, the set of 𝐻 ′ -weights given byforms an arithmetic sequence 𝑎, 𝑎 + 𝑑, … , 𝑎 + (𝑡 − 1)𝑑 where 𝑎 > 0, 𝑑 ⩾ 0 are two fixed integers and 𝑡 is the number of all subgraphs of 𝐺 isomorphic to 𝐻. Moreover, such a labeling 𝜑 is called super if the smallest possible labels appear on the vertices. A (super) (𝑎, 𝑑)-𝐻-antimagic graph is a graph that admits a (super) (𝑎, 𝑑)-𝐻-antimagic total labeling. In this paper the existence of super (𝑎, 𝑑)-𝐻-antimagic total labelings for the 𝑚-shadow and the closed 𝑚-shadow of a connected 𝐺 for several values of 𝑑 is proved.Problem 1. For 2 ⩽ 𝑡 ⩽ 𝑚 − 1, determine 𝐷 𝑡 (𝐺)-supermagic labelings for the 𝑚-shadow 𝐷 𝑚 (𝐺) of any connected graph 𝐺 on 𝑝 vertices and 𝑞 edges.

Combined in vitro/in silico approaches, molecular dynamics simulations and safety assessment of the multifunctional properties of thymol and carvacrol: A comparative insight

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Structural Performance of Additively Manufactured Composite Lattice Structures: Strain Rate, Cell Geometry, and Weight Ratio Effects

“…[ 1–5 ] Additive thinking is indeed infiltrating the scientific and technological communities, racing to realize the benefits of these empowering advanced manufacturing techniques in every industrial domain, from consumer goods to nutrient alternatives and extreme engineering such as space travel. [ 6–9 ] For example, sporting equipment, including protective helmets and running shoes, have been mass‐produced using various additive manufacturing approaches, [ 6,7,10 ] enabling their retail to the general public at competitive prices. In short, the premises of additive manufacturing, such as broadening the design envelope, just‐in‐time delivery, and efficient production, are manifested in real‐life applications and facilitate the rapid translation of products from the bench to the market.…”

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“…Abbas et al (2023) introduced a new ligand called DNP and its chelates, showing promising antibacterial activity against Gram‐positive and Gram‐negative bacteria 7 . Alkhatib and Alsulami (2023) synthesized a Schiff base ligand known as HL 7 . They complexed it with cobalt (II), nickel (II), and copper (II) acetate, finding that the Ni (II) complex displayed significant inhibition of WISH cell growth, indicating its potential as an effective anticancer agent.…”

“…These approaches enhanced our understanding of the compounds' structural properties and mechanisms of action. Overall, metal complexes contribute to addressing pressing issues related to antibiotic resistance and the need for more effective anticancer agents 7,8 . Metal‐based complexes of Ru (II/III) have attracted great attention as alternative anticancer agents because of their unique interactions with nucleic acids such as DNA 9 .…”