On Non Inclusive Distance Vertex Irregularity Strength of Tadpole and Path Corona Path Graphs

Muhammad Bilal, Diari Indriati, Vika Yugi Kurniawan

Abstract

Let 𝐺 = (𝑉, 𝐸) be a connected and simple graph with vertex set 𝑉(𝐺) and edge set
𝐸(𝐺). A non inclusive distance vertex irregular labeling of a graph 𝐺 is a mapping of πœ† ∢
(𝑉, 𝐺) β†’ {1, 2, … , π‘˜} such that the weights calculated for all vertices are distinct. The weight of a vertex 𝑣, under labeling πœ†, denoted by 𝑀𝑑(𝑣), is defined as the sum of the label of all vertices adjacent to 𝑣 (distance 1 from 𝑣). A non inclusive distance vertex irregularity strength of graph 𝐺, denoted by 𝑑𝑖𝑠(𝐺), is the minimum value of the largest label π‘˜ over all such non inclusive distance vertex irregular labeling. In this research, we determined 𝑑𝑖𝑠(𝐺) from π‘‡π‘š,𝑛 graph with π‘š β‰₯ 3, π‘š odd, π‘Žπ‘›π‘‘ 𝑛 β‰₯ 1 and 𝑃𝑛 βŠ™ 𝑃𝑛 graph π‘€π‘–π‘‘β„Ž 𝑛 β‰₯ 2 and 𝑛 even.

Keywords

non inclusive distance irregular labeling, non inclusive distance vertex irregularity strength, tadpole graph, path corona path graph

References

Bača, M., S. Jendrol’, M. Miller, and J. Ryan, (2007) On Irregular Total Labeling, Discrete Mathematics 307, 1378-1388.

Bong, N.H. and L. Yuqing, (2017) On Distance-Irregular Labelings of Cycles and Wheels, Australasian Journal of Combinatorics 69(3), 315-322.

Chartrand, G., M.S. Jacobson, J. Lehel, O.R. Oellerman, S. Ruiz, and F. Saba, (1998),

Irregular Networks, Congr. Numer. 64, 187-192.

Fruncth and F. Harary, (1970), On The Corona of Two Graphs, Aequationes Math 4, 322325.

Gallian, J.A., A Dynamic Survey of Graph Labeling, (2017), The Electronic Journal of Combinatorics, 20, (6), 1-432.

Kotzig, A. and A. Rosa, (1970), Magic Valuations of Finite Graphs, Canad. Math. Bull. 13, 451-461.

MacDougall, J.A., M. Miller, Slamin, and W.D. Wallis, (2002), Vertex-Magic TotalLabelings of Graphs, Util. Math. 61, 3-21.

Miller, M., C. Rodger, and R. Simanjuntak, (2003), Distance Magic Labelings of Graphs, Aurstralas. J. Combin. 28, 305-315.

SedlÑčk, J., (1963), Theory of Graphs and Its Applications, House Czechoslovak Acad. Sci. Prague, 163-164.

Slamin, (2017), On Distance Irregular Labelling of Graphs, Far East Journal of

Mathematical Sciences 102, 919-932.

Wallis, W.D., 2001, Magic Graph, BirkhΓ€user, Basel, Berlin,

Refbacks

  • There are currently no refbacks.

Comments on this article

View all comments