In 2007, Baker and Norine proved an analogue of the Riemann-Roch Theorem for finite graphs. Motivated by the Weierstrass semigroup of a point on a nonsingular projective curve X, it is natural to consider analogues for a vertex P on a finite, connected graph which has no loops. Let Hr(P ) be the set of nonnegative integers α such that r(αP ) = r((α − 1)P ) + 1, wherer(D) denotes the dimension o...

Let R be a commutative ring with unity. The set Z(R) of zero-divisors in a ring does not possess any obvious algebraic structure; consequently, the study of this set has often involved techniques and ideas from outside algebra. Several recent attempts, among them [2], [3] have focused on studying the so-called zero-divisor graph ΓR, whose vertices are the zero-divisors of R, with xy being an ed...

In this article, a log del Pezzo surface of index two means a projective normal non-Gorenstein surface S such that (S, 0) is a log-terminal pair, the anti-canonical divisor −KS is ample and that 2KS is Cartier. The log del Pezzo surfaces of index two are shown to be constructed from data (X,E,∆) called fundamental triplets consisting of a non-singular rational surface X, a simple normal crossin...

In this work and its sequel, we study the expanding phenomenon of matrices over a finite chain ring large residue field. A sum-product estimate is proved. It shown that [Formula: see text] moderate expander on with exponent text]. These results generalize main theorems in recent [29] Xie Ge. The proofs use spectral graph theory elementary divisor theory.

An edge-coloring of a graph G with colors 1, . . . , t is an interval t-coloring if all colors are used, and the colors of edges incident to each vertex of G are distinct and form an interval of integers. A graph G is interval colorable if it has an interval t-coloring for some positive integer t. In this note we prove that K1,m,n is interval colorable if and only if gcd(m+ 1, n+ 1) = 1, where ...

Let R be a fnite commutative ring and N(R) be the set of non unit elements of R. The non unit graph of R, denoted by Gamma(R), is the graph obtained by setting all the elements of N(R) to be the vertices and defning distinct vertices x and y to be adjacent if and only if x - yin N(R). In this paper, the basic properties of Gamma(R) are investigated and some characterization results regarding co...