2014
DOI: 10.1007/s10878-014-9775-4
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$$L(j,k)$$ L ( j , k ) -labeling number of Cartesian product of path and cycle

Abstract: For positive numbers j and k, an L( j, k)-labeling f of G is an assignment of numbers to vertices of G such thatThe span of f is the difference between the maximum and the minimum numbers assigned by f . The L( j, k)-labeling number of G, denoted by λ j,k (G), is the minimum span over all L( j, k)-labelings of G. In this article, we completely determine the L( j, k)-labeling number (2 j ≤ k) of the Cartesian product of path and cycle.

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Cited by 24 publications
(23 citation statements)
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“…Silicene is a silicon-based material which has only recently been experimentally-realized [1,2]. Like carbon-based graphene [3], it has a two-dimensional (2D) structure which gives rise to its unusual and potentially important properties.…”
Section: Introductionmentioning
confidence: 99%
“…Silicene is a silicon-based material which has only recently been experimentally-realized [1,2]. Like carbon-based graphene [3], it has a two-dimensional (2D) structure which gives rise to its unusual and potentially important properties.…”
Section: Introductionmentioning
confidence: 99%
“…PANI shows a vibrational peak at 633.98 cm À1 due to the stretching vibration of C-H. [39] The bending vibrational peak of the aromatic C-H bond in the 1,4-disubstituted aromatic ring of protonated emeraldine corresponds to 821.5 and 1161 cm À1 . [40][41][42] The band at 1248.6 cm À1 is the stretching peaks of C-N. [38,40] The band at 1314.7 cm À1 is the π-electron delocalization of the polymer and/or with the C═N-C stretching vibration. [42] The peaks at 1502.2 and 1616.5 cm À1 related to C═C stretching vibrations.…”
Section: Resultsmentioning
confidence: 99%
“…[40][41][42] The band at 1248.6 cm À1 is the stretching peaks of C-N. [38,40] The band at 1314.7 cm À1 is the π-electron delocalization of the polymer and/or with the C═N-C stretching vibration. [42] The peaks at 1502.2 and 1616.5 cm À1 related to C═C stretching vibrations. [43] The peak at 2158.9 cm À1 represents N═C═O.…”
Section: Resultsmentioning
confidence: 99%
“…Rhamnolipids FTIR, HPLC, TLC [57,58] Sophorolipids FTIR,HPLC [59] Trehalolipids HPLC [60] Surfactin HPLC [61] Phospholipids GC/MS [62] …”
Section: Biosurfactant Analytical Methods Referencementioning
confidence: 99%