DISTANCE SPECTRUM OF TWO FAMILIES OF GRAPHS
1
조회
0
다운로드
0
인용
0
참고문헌
키워드
초록
Let H₁ and H₂ be two copies of the complete graph Kₙ, n ≥ 3 with vertex sets V(H₁) = {v₁, v₂, ..., vₙ} and V(H₂) = {u₁, u₂, ..., uₙ}. Graph Γ(n, p), 1 ≤ p ≤ n−1, is obtained from the union of graphs H₁ and H₂ by adding edges {uᵢvᵢ | i ∈ {1, 2, ..., p}}. Graph Θ(n) is obtained from the union of graphs H₁ and H₂ by joining each vertex vᵢ of H₁ to every vertex in {u₁, u₂, ..., uₙ} \ {uᵢ}, for i = 1, 2, ..., n. The adjacency spectrum of Γ(n, p) and Θ(n) were determined in [9]. An open problem posed in [7] was to find families of graphs of diameter greater than two, for which the adjacency and distance spectrum are both integral. To answer the open problem, the distance spectrum of the above family of graphs is calculated, and new distance equienergetic graphs are constructed in this paper.
Abstract
Let H₁ and H₂ be two copies of the complete graph Kₙ, n ≥ 3 with vertex sets V(H₁) = {v₁, v₂, ..., vₙ} and V(H₂) = {u₁, u₂, ..., uₙ}. Graph Γ(n, p), 1 ≤ p ≤ n−1, is obtained from the union of graphs H₁ and H₂ by adding edges {uᵢvᵢ | i ∈ {1, 2, ..., p}}. Graph Θ(n) is obtained from the union of graphs H₁ and H₂ by joining each vertex vᵢ of H₁ to every vertex in {u₁, u₂, ..., uₙ} \ {uᵢ}, for i = 1, 2, ..., n. The adjacency spectrum of Γ(n, p) and Θ(n) were determined in [9]. An open problem posed in [7] was to find families of graphs of diameter greater than two, for which the adjacency and distance spectrum are both integral. To answer the open problem, the distance spectrum of the above family of graphs is calculated, and new distance equienergetic graphs are constructed in this paper.
참고문헌 (14)
- L. Collatz, "Spektren endlicher Grafen", Abhandlungen aus dem Mathematischen Seminar der Universitat Hamburg, 21, pp.63~77 (1957)
- D. M. Cvetkovic, "Spectra of graphs. Theory and Applications" (1980), Academic Press
- P. J. Davis, "Circulant matrices" (1979)
- R. L. Graham, "On the addressing problem for loop switching", Bell Syst. Tech. J, 50, pp.2495~2519 (1971)
- Gopalapillai Indulal, "D-spectrum and D-energy of Complements of Iterated Line Graphs of Regular Graphs", Algebraic Structures and Their Applications, 4(1), pp.53~58 (2017)
- I. Gutman, "The energy of a graph", Ber. Math. Stat. Sekt. Forschungsz. Graz, 103, pp.1~22 (1978)
- Gopalapillai Indulal, "Distance spectrum of Indu–Bala product of graphs", AKCE International Journal of Graphs and Combinatorics, 13, pp.230~234 (2016)
- Gopalapillai Indulal, "On distance energy of graphs", Match Commun. Math. Comput. Chem, 60, pp.461~472 (2008)
- Ivan Stankovic, "Small and not so Small Equienergetic Graphs", MATCH Commun. Math. Comput. Chem, 61, pp.443~450 (2009)
- M. Liu, "A note on D-equienergetic graphs", MATCH Commun. Math. Comput. Chem, 64, pp.135~140 (2010)
- H. S. Ramane, "Distance Equienergetic Graphs", MATCH Commun. Math. Comput. Chem, 60, pp.473~484 (2008)
- Mustapha Aouchiche, "Distance spectra of graphs: A survey", Linear Algebra and its Applications, 458, pp.301~386 (2014)
- D. Stevanovic, "Distance in Molecular Graphs—Theory", pp.139~176 (2012), University of Kragujevac
- D. Stevanovic, "The distance spectrum and energy of the compositions of regular graphs", Applied Mathematics Letters, 22, pp.1136~1140 (2009)