Induced Matchings and the v-Number of Graded Ideals

We give a formula for the v-number of a graded ideal that can be used to compute this number. Then, we show that for the edge ideal I(G) of a graph G, the induced matching number of G is an upper bound for the v-number of I(G) when G is very well-covered, or G has a simplicial partition, or G is well-covered connected and contains neither four, nor five cycles. In all these cases, the v-number of I(G) is a lower bound for the regularity of the edge ring of G. We classify when the induced matching number of G is an upper bound for the v-number of I(G) when G is a cycle and classify when all vertices of a graph are shedding vertices to gain insight into the family of W2-graphs.

WDM Metro Networks

With increasing traffic, legacy SONET/SDH-metro rings have mostly been upgraded using point-to-point WDM (PPWDM) transmission. For cost-effective realization, WDM metro networks also employ wavelength-routed optical networking (WRON), where a wavelength is bypassed optically at intermediate nodes. Metro networks are generally split into core and edge rings, with the metro-core ring interfacing with the long-haul backbone and the metro-edge ring interconnecting the metro-core ring with the access segment. The PPWDM/WRON transmission is employed with each wavelength using circuit-switched SONET/SDH transmission, though the metro-edge rings can also use packet-switching to enhance bandwidth utilization with bursty traffic from the access segment. In this chapter, we consider first WDM metro networks using PPWDM/WRON-based rings and present their design methodologies using LP-based and heuristic schemes. We describe some packet-switched WDM ring testbeds and examine the possible improvement in bandwidth utilization therein, as compared to the circuit-switched WRON rings. (143 words)

Edge rings of bipartite graphs with linear resolutions

On the one hand, Ohsugi and Hibi characterized the edge ring of a finite connected simple graph with a 2-linear resolution. On the other hand, Hibi, Matsuda and the author conjectured that the edge ring of a finite connected simple graph with a [Formula: see text]-linear resolution, where [Formula: see text], is a hypersurface and proved the case [Formula: see text]. In this paper, we solve this conjecture for the case of finite connected simple bipartite graphs.

The Regularity of Edge Rings and Matching Numbers

Let K [ G ] denote the edge ring of a finite connected simple graph G on [ d ] and mat ( G ) the matching number of G. It is shown that reg ( K [ G ] ) ≤ mat ( G ) if G is non-bipartite and K [ G ] is normal, and that reg ( K [ G ] ) ≤ mat ( G ) − 1 if G is bipartite.

Cable Nets with Elastically Deformable Edge Ring

The objective of this work is to present the behavior of prestressed cable net structures anchored to an elastically deformable edge ring and to propose a methodology for their preliminary analysis and design. The interaction between the edge ring and the cable net is examined. The proposed method for preliminary analysis and design is based on charts, produced via geometrically nonlinear analysis for a structure with specific geometrical and mechanical characteristics, and suitable transformation relations. Using them, it is possible to calculate the maximum tension and deflection of the cables and the maximum internal forces, moments and deformations of the edge ring, for any cable net with similar geometry. The charts presented in this paper cover nets with circular plan view. Nevertheless, the methodology can be readily extended to other shapes.