The Italian Domination Numbers of Generalized Petersen Graphs P(n,3)

Gao, Hong; Xi, Changqing; Li, Kun; Zhang, Qingfang; Yang, Yuansheng

doi:10.3390/math7080714

Cited by 10 publications

(5 citation statements)

References 17 publications

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“…A facile seed-mediated synthesis method for the fabrication of spiked AgNSs, using a two-step CM reduction method, was employed for the SERS-based detection of Congo red and probenecid molecules [29,30]. Multi-branched three-dimensional (3D) silver nanostructures on flexible supports, prepared by the electrochemical deposition technique, have shown improved signal uniformity and 10 −9 M LOD [31].…”

Section: Introductionmentioning

confidence: 99%

Multi-spiked silver stars for ultrasensitive and multiplexed SERS detection of analytes

Verma¹,

Soni²

2021

J. Phys. D: Appl. Phys.

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We report a facile synthesis procedure of multi-spiked star-shaped silver structures with sharp morphological features and broad plasmonic absorption for the fabrication of surface-enhanced Raman scattering (SERS) substrates. The multi-branched morphology of silver stars (AgSs) provides copious hot spots on SERS-active substrates to detect ultra-low concentrations of probe molecules. Significant Raman intensity enhancement factors of the order of ∼10 9 were obtained for the different analytes. The AgSs-based SERS-substrates show ultra-sensitive SERS performance for the detection of probe molecules rhodamine B, methylene blue, thiram, and phosmet with detectabilities 2.6 × 10 −16 , 1.97 × 10 −17 , 1.78 × 10 −18 , and 4.08 × 10 −14 M concentration, respectively. The improved SERS sensitivity originated from the additional contribution of chemical enhancement, corroborated by the appearance of new Raman bands in the SERS spectrum. For practical applications, SERS based molecule detection requires high sensitivity, specificity, uniformity, and reproducibility of the SERS signal. We demonstrate the simultaneous detection of multiple analytes, with high specificity, from the complex mixture of molecules with different compositions. The fabricated SERS substrates provide excellent signal uniformity and reproducibility with ∼10% relative standard deviation for the multiple substrates. The real-life application of AgSs-based SERS substrates with the spiked apple juice and ITO-coated PET films without pre-processing is also demonstrated.

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Section: Introductionmentioning

confidence: 99%

Multi-spiked silver stars for ultrasensitive and multiplexed SERS detection of analytes

Verma¹,

Soni²

2021

J. Phys. D: Appl. Phys.

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“…Recently, Chellali et al [6] have introduced the Roman {2}-dominating function f as follows. A Roman {2}-dominating function is a function f : V → {0, 1, 2} such that for every vertex v ∈ V, with f (v) = 0, f (N(v)) ≥ 2 where f (N(v)) = ∑ x∈N(v) f (x), that is, either v has a neighbor u with f (u) = 2, or has two neighbors x, y with f (x) = f (y) = 1 [7].…”

Section: Introductionmentioning

confidence: 99%

Total Roman {3}-domination in Graphs

2020

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For a graph G = ( V , E ) with vertex set V = V ( G ) and edge set E = E ( G ) , a Roman { 3 } -dominating function (R { 3 } -DF) is a function f : V ( G ) → { 0 , 1 , 2 , 3 } having the property that ∑ u ∈ N G ( v ) f ( u ) ≥ 3 , if f ( v ) = 0 , and ∑ u ∈ N G ( v ) f ( u ) ≥ 2 , if f ( v ) = 1 for any vertex v ∈ V ( G ) . The weight of a Roman { 3 } -dominating function f is the sum f ( V ) = ∑ v ∈ V ( G ) f ( v ) and the minimum weight of a Roman { 3 } -dominating function on G is the Roman { 3 } -domination number of G, denoted by γ { R 3 } ( G ) . Let G be a graph with no isolated vertices. The total Roman { 3 } -dominating function on G is an R { 3 } -DF f on G with the additional property that every vertex v ∈ V with f ( v ) ≠ 0 has a neighbor w with f ( w ) ≠ 0 . The minimum weight of a total Roman { 3 } -dominating function on G, is called the total Roman { 3 } -domination number denoted by γ t { R 3 } ( G ) . We initiate the study of total Roman { 3 } -domination and show its relationship to other domination parameters. We present an upper bound on the total Roman { 3 } -domination number of a connected graph G in terms of the order of G and characterize the graphs attaining this bound. Finally, we investigate the complexity of total Roman { 3 } -domination for bipartite graphs.

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“…In other words, we say that a vertex v for which f (v) ∈ {1, 2} dominates itself, while a vertex v with f (v) = 0 is dominated by f if it has at least two in-neighbors assigned 1 under f or one in-neighbor w with f (w) = 2. An Italian dominating function f : The Italian dominating functions in graphs and digraphs have been studied in [1][2][3][4][5][6][7]. The authors of [2] introduce the concept of Italian domination and give bounds, relating the Italian domination number to some other domination parameters.…”

Section: Introductionmentioning

confidence: 99%