17,084 research outputs found
Flow-up Bases for Generalized Spline Modules on Arbitrary Graphs
Let R be a commutative ring with identity. An edge labeled graph is a graph
with edges labeled by ideals of R. A generalized spline over an edge labeled
graph is a vertex labeling by elements of R, such that the labels of any two
adjacent vertices agree modulo the label associated to the edge connecting
them. The set of generalized splines forms a subring and module over R. Such a
module it is called a generalized spline module. We show the existence of a
flow-up basis for the generalized spline module on an edge labeled graph over a
principal ideal domain by using a new method based on trails of the graph. We
also give an algorithm to determine flow-up bases on arbitrary ordered cycles
over any principal ideal domain
The total zero-divisor graph of commutative rings
In this paper we initiate the study of the total zero-divisor graphs over
commutative rings with unity. These graphs are constructed by both relations
that arise from the zero-divisor graph and from the total graph of a ring. We
characterize Artinian rings with the connected total zero-divisor graphs and
give their diameters. Moreover, we compute major characteristics of the total
zero-divisor graphs of the ring of integers modulo and
prove that the total zero-divisor graphs of and
are isomorphic if and only if
Independent sets of some graphs associated to commutative rings
Let be a simple graph. A set is independent set of
, if no two vertices of are adjacent. The independence number
is the size of a maximum independent set in the graph. %An
independent set with cardinality Let be a commutative ring with nonzero
identity and an ideal of . The zero-divisor graph of , denoted by
, is an undirected graph whose vertices are the nonzero
zero-divisors of and two distinct vertices and are adjacent if and
only if . Also the ideal-based zero-divisor graph of , denoted by
, is the graph which vertices are the set {x\in R\backslash I |
xy\in I \quad for some \quad y\in R\backslash I\} and two distinct vertices
and are adjacent if and only if . In this paper we study the
independent sets and the independence number of and .Comment: 27 pages. 22 figure
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