Let R be a commutative ring with identity and A(R) be the set of ideals with nonzero annihilator. The annihilating-ideal graph of R is defined as the graph AG(R) with the vertex set A(R) = A(R)r {0} and two distinct vertices I and J are adjacent if and only if IJ = 0. In this paper, we study the domination number of AG(R) and some connections between the domination numbers of annihilating-ideal...

Let JG denote the binomial edge ideal of a connected undirected graph on n vertices. This is the ideal generated by the binomials xiyj − xjyi, 1 ≤ i < j ≤ n, in the polynomial ring S = K[x1, . . . , xn, y1, . . . , yn] where {i, j} is an edge of G. We study the arithmetic properties of S/JG for G, the complete bipartite graph. In particular we compute dimensions, depths, Castelnuovo-Mumford reg...

Let X be an m × n matrix of indeterminates, m ≤ n, and T a new indeterminate. Consider the polynomial rings R0 = K[X] and R = R0[T ]. For a given positive integer t ≤ m, consider the ideal It = It(X) generated by the t-minors (i. e. the determinants of the t× t submatrices) of X . Using all these determinantal ideals, we define a new ideal J in R = R0[T ], which we call the generic graph constr...

This paper explores finding the number nh of undirected hamiltonian paths in an undirected graph G = (V,E) using lumped/ideal circuits, specifically low-pass filters. Ideal analog computation allows one to computer nh in a short period of time, but in practice, precision problems disturb this ideal nature. A digital/algorithmic approach is proposed, and then it is shown that the approach/method...

We show that attaching a whisker (or pendant) at the vertices of cycle cover graph results in new with following property: all symbolic powers its ideal are Koszul or, equivalently, componentwise linear. This extends previous work where whiskers were added to or vertex graph.

We study representations of ideal languages by means of strongly connected synchronizing automata. For every finitely generated ideal language L we construct such an automaton with at most 2 states, where n is the maximal length of words in L. Our constructions are based on the De Bruijn graph.

We study the initial ideal of binomial edge ideal in degree 2 ([in<(JG)]2), associated to a graph G. We computed dimension, depth, Castelnuovo-Mumford regularity, Hilbert function and Betti numbers of [in<(JG)]2 for some classes of graphs. AMS Mathematics Subject Classification (2010): 05E40, 16E30

We introduce compactness theorems for generalized colorings and derive several particular compactness theorems from them. It is proved that the theorems and many of their consequences are equivalent in ZF set theory to BPI, the Prime Ideal Theorem for Boolean algebras. keywords: generalized graph colorings, compactness, prime ideal theorem MSC: 05C15; 03E25 .

We consider the problem of finding edge search strategies of minimum cost. The cost of a search strategy is the sum of searchers used in the clearing steps of the search. One of the natural questions is whether it is possible to find a search strategy that minimizes both the cost and the number of searchers used to clear a given graph G. We call such a strategy ideal. We prove, by an example, t...