Toroidal and Projective Cyclic Graphs (original) (raw)
On the enumeration of a class of toroidal graphs
2018
We present enumerations of a class of toroidal graphs which give rise to semi-equivelar maps. There are eleven different types of semi-equivelar maps on the torus. These are of the types 36\{3^{6}\}36, 44\{4^{4}\}44, 63\{6^{3}\}63, 33,42\{3^{3}, 4^{2}\}33,42, 32,4,3,4\{3^{2}, 4, 3, 4\}32,4,3,4, 3,6,3,6\{3, 6, 3, 6\}3,6,3,6, 34,6\{3^{4}, 6\}34,6, 4,82\{4, 8^{2}\}4,82, 3,122\{3, 12^{2}\}3,122, 4,6,12\{4, 6, 12\}4,6,12, 3,4,6,4\{3, 4, 6, 4\}3,4,6,4. We know the classification of the maps of types 36\{3^{6}\}36, 44\{4^{4}\}44, 63\{6^{3}\}63 on the torus. In this article, we attempt to classify maps of types 33,42\{3^{3}, 4^{2}\}33,42, 32,4,3,4\{3^{2}, 4, 3, 4\}32,4,3,4, 3,6,3,6\{3, 6, 3, 6\}3,6,3,6, 34,6\{3^{4}, 6\}34,6, 4,82\{4, 8^{2}\}4,82, 3,122\{3, 12^{2}\}3,122, 4,6,12\{4, 6, 12\}4,6,12, 3,4,6,4\{3, 4, 6, 4\}3,4,6,4 on the torus.
On enumeration of a class of toroidal graphs
2013
We present enumerations of a class of toroidal graphs which give rise to semi-equivelar maps. There are eleven different types of semi-equivelar maps on the torus. These are of the types {3^6}, {4^4}, {6^3}, {3^3, 4^2}, {3^2, 4, 3, 4}, {3, 6, 3, 6}, {3^4, 6}, {4, 8^2}, {3, 12^2}, {4, 6, 12}, {3, 4, 6, 4}. We know the classification of the maps of types {3^6}, {4^4}, {6^3} on the torus. In this article, we attempt to classify maps of types {3^3, 4^2}, {3^2, 4, 3, 4}, {3, 6, 3, 6}, {3^4, 6}, {4, 8^2}, {3, 12^2}, {4, 6, 12}, {3, 4, 6, 4} on the torus.
Classification of Rings with Toroidal and Projective Coannihilator Graph
Journal of Mathematics
Let S be a commutative ring with unity, and a set of nonunit elements is denoted by W S . The coannihilator graph of S , denoted by A G ′ S , is an undirected graph with vertex set W S ∗ (set of all nonzero nonunit elements of S ), and α ∼ β is an edge of A G ′ S ⇔ α ∉ α β S or β ∉ α β S , where δ S denotes the principal ideal generated by δ ∈ S . In this study, we first classify finite ring S , for which A G ′ S is isomorphic to some well-known graph. Then, we characterized the finite ring S , for which A G ′ S is toroidal or projective.
Generator graphs for cyclic groups
THE 11TH NATIONAL CONFERENCE ON MATHEMATICAL TECHNIQUES AND APPLICATIONS, 2019
The present paper is focusing on the connection between the cyclic groups and graphs with cycles especially containing K 3 graphs. We established few results on the number of K 3 graphs in a cyclic group with respect to the generators that are generating the cyclic groups. We proved the results depending on the order of different cyclic groups.
The obstructions for toroidal graphs with no K3,3’s
2005
Forbidden minors and subdivisions for toroidal graphs are numerous. We consider the toroidal graphs with no K3,3-subdivisions that coincide with the toroidal graphs with no K3,3-minors. These graphs admit a unique decomposition into planar components and have short lists of obstructions. We provide the complete lists of four forbidden minors and eleven forbidden subdivisions for the toroidal graphs with no K3,3’s and prove that the lists are sufficient.
Abstract: In this paper we introduced a new concept of graph of any finite group and we obtained graphs of some finite groups. Moreover some results on this concept are proved. Keywords: Group, Abelian group, Cyclic group, Graph, Degree of a graph.