The Riemann-Roch theorem on a graph G is closely related to Alexander duality in combinatorial commutive algebra. We study the lattice ideal given by chip firing on G and the initial ideal whose standard monomials are the G-parking functions. When G is a saturated graph, these ideals are generic and the Scarf complex is a minimal free resolution. Otherwise, syzygies are obtained by degeneration...

We characterize all graphs whose binomial edge ideals have a linear resolution. Indeed, we show that complete graphs are the only graphs with this property. We also compute some graded components of the first Betti number of the binomial edge ideal of a graph with respect to the graphical terms. Finally, we give an upper bound for the Castelnuovo-Mumford regularity of the binomial edge ideal of...

let $r$ be a commutative ring with zero-divisor set $z(r)$. the total graph of $r$, denoted by$t(gamma(r))$, is the simple (undirected) graph with vertex set  $r$ where two distinct vertices areadjacent if their sum lies in $z(r)$.this work considers minimum zero-sum $k$-flows for $t(gamma(r))$.both for $vert rvert$ even and the case when $vert rvert$ is odd and $z(g)$ is an ideal of $r$it is s...

Double orthogonality in the set of eigenvectors of any symmetric graph matrix is exploited to propose a set of nodal centrality metrics, that is “ideal” in the sense of being complete, uncorrelated and mathematically precisely defined and computable. Moreover, we show that, for each node m, such a nodal eigenvector centrality metric reflects the impact of the removal of node m from the graph at...

Let R be a commutative ring with nonzero identity and let I be an ideal of R. The zero-divisor graph of R with respect to I, denoted by ΓI(R), is the graph whose vertices are the set {x ∈ R \ I| xy ∈ I for some y ∈ R \ I} with distinct vertices x and y adjacent if and only if xy ∈ I. In the case I = 0, Γ0(R), denoted by Γ(R), is the zero-divisor graph which has well known results in the literat...

At the cost of restricting the nature of the involved K-groups, we prove a classification result for a hitherto unexplored class of graph C∗-algebras, allowing us to classify all graph C∗-algebras on finitely many vertices with a finite linear ideal lattice if all pair of vertices are connected by infinitely many edges when they are connected at all. 2000 Mathematics Subject Classification: Pri...

Let G be a simple graph on d vertices. We define a monomial ideal K in the Stanley-Reisner ring A of the order complex of the Boolean algebra on d atoms. The monomials in K are in one-to-one correspondence with the proper colorings of G. In particular, the Hilbert polynomial of K equals the chromatic polynomial of G. The ideal K is generated by square-free monomials, so A/K is the Stanley-Reisn...

We study the ideal triangulation graph T (S) of a punctured surface S of finite type. We show that if S is not the sphere with at most three punctures or the torus with one puncture, then the natural map from the extended mapping class group of S into the simplicial automorphism group of T (S) is an isomorphism. We also show that under the same conditions on S, the graph T (S) equipped with its...