When a line graph associated to annihilating-ideal graph of a lattice is planar or projective
- 33 Downloads
Let (L,∧, ∨) be a finite lattice with a least element 0. AG(L) is an annihilating-ideal graph of L in which the vertex set is the set of all nontrivial ideals of L, and two distinct vertices I and J are adjacent if and only if I ∧ J = 0. We completely characterize all finite lattices L whose line graph associated to an annihilating-ideal graph, denoted by L(AG(L)), is a planar or projective graph.
Keywordsannihilating-ideal graph lattice line graph planar graph projective graph
MSC 201005C75 05C10 06B10
Unable to display preview. Download preview PDF.
- D. F. Anderson, M. C. Axtell, J. A. Stickles: Zero-divisor graphs in commutative rings. Commutative Algebra, Noetherian and Non-Noetherian Perspectives (M. Fontana et al., eds.). Springer, New York, 2011, pp. 23–45.Google Scholar
- H. H. Glover, J. P. Huneke, C. S. Wang: 103 graphs that are irreducible for the projective plane. J. Comb. Theory, Ser. B 27 (1979), 332–370.Google Scholar
- C. Godsil, G. Royle: Algebraic Graph Theory. Graduate Texts in Mathematics 207, Springer, New York, 2001.Google Scholar
- W. S. Massey: Algebraic Topology: An Introduction. Graduate Texts in Mathematics 56, Springer, New York, 1977.Google Scholar
- J. B. Nation: Notes on Lattice Theory. 1991–2009. Available at http: //www.math. hawaii.edu/~jb/books.html.Google Scholar
- G. Ringel: Map Color Theorem. Die Grundlehren der mathematischen Wissenschaften 209, Springer, Berlin, 1974.Google Scholar
- J. Sedláček: Some properties of interchange graphs. Theory Graphs Appl. Proc. Symp. Smolenice, 1963, Czechoslovak Acad. Sci., Praha, 1964, pp. 145–150.Google Scholar
- A. T. White: Graphs, Groups and Surfaces. North-Holland Mathematics Studies 8, North-Holland Publishing, Amsterdam-London; American Elsevier Publishing, New York, 1973.Google Scholar