Let R be a commutative ring with nonzero identity and J(R) the Jacobson radical of R. The Jacobson graph of R, denoted by JR, is defined as the graph with vertex set RJ(R) such that two distinct vertices x and y are adjacent if and only if 1 − xy is not a unit of R. The genus of a simple graph G is the smallest nonnegative integer n such that G can be embedded into an orientable surface Sn. In this paper, we investigate the genus number of the compact Riemann surface in which JR can be embedded and explicitly determine all finite commutative rings R (up to isomorphism) such that JR is toroidal., Krishnan Selvakumar, Manoharan Subajini., and Obsahuje seznam literatury
We study improper interval edge colourings, defined by the requirement that the edge colours around each vertex form an integer interval. For the corresponding chromatic invariant (being the maximum number of colours in such a colouring), we present upper and lower bounds and discuss their qualities; also, we determine its values and estimates for graphs of various families, like wheels, prisms or complete graphs. The study of this parameter was inspired by the interval colouring, introduced by Asratian, Kamalian (1987). The difference is that we relax the requirement on the original colouring to be proper., Peter Hudák, František Kardoš, Tomáš Madaras, Michaela Vrbjarová., and Obsahuje seznam literatury
Let G be a finite group. The intersection graph ΔG of G is an undirected graph without loops and multiple edges defined as follows: the vertex set is the set of all proper nontrivial subgroups of G, and two distinct vertices X and Y are adjacent if X ∩ Y ≠ 1, where 1 denotes the trivial subgroup of order 1. A question was posed by Shen (2010) whether the diameters of intersection graphs of finite non-abelian simple groups have an upper bound. We answer the question and show that the diameters of intersection graphs of finite non-abelian simple groups have an upper bound 28. In particular, the intersection graph of a finite non-abelian simple group is connected., Xuanlong Ma., and Obsahuje seznam literatury