This paper introduces logarithmic curvature and torsion graphs for analyzing planar and space curves. We present a method for drawing these graphs from any differentiable parametric curves and clarify the characteristics of these graphs. We show several examples of theses graphs drawn from planar and 3D Bézier curves. From the graphs, we can see some interesting properties of curves that cannot...

Let E be a Q-isogeny class of elliptic curves defined over Q. The isogeny graph associated to is which has vertex for each element and an edge prime degree that maps one another E, with the recorded as label edge. isogeny-torsion where, in addition, we abstract group structure torsion subgroup Q corresponding curve. main result article determination graphs classes correspond infinitely many j-i...

Magnitude homology is a bigraded theory for finite graphs defined by Hepworth and Willerton, categorifying the power series invariant known as magnitude which was introduced Leinster. We analyze structure implications of torsion in homology. show that any finitely generated abelian group may appear subgroup graph, and, particular, given prime order can graph there are infinitely many such graph...

A short proof, using graphs and groupoids, is given of Brodskĭı’s theorem that torsion-free one-relator groups are locally indicable.

In this paper, we investigate the strength of chromatic symmetric homology as a graph invariant. Chromatic is lift function for graphs to homological setting, and its Frobenius characteristic q,t generalization function. We exhibit three pairs where each pair has same but distinct over C Sn-modules. also show that integral contains torsion, based on computations, conjecture Z2-torsion in bigrad...

We show that Cayley graphs of finitely generated Abelian groups are rather rigid. As a consequence we obtain that two finitely generated Abelian groups admit isomorphic Cayley graphs if and only if they have the same rank and their torsion parts have the same cardinality. The proof uses only elementary arguments and is formulated in a geometric language.

In recent years, discrete spaces such as graphs attract much attention as models for physical spacetime or as models for testing the spirit of non-commutative geometry. In this work, we construct the differential algebras for graphs by extending the work of Dimakis et al and discuss linear connections and curvatures on graphs. Especially, we calculate connections and curvatures explicitly for t...

J.L. Andersen proved that there is 5-torsion in the bottom nonvanishing homology group of the simplicial complex of graphs of degree at most two on seven vertices. We use this result to demonstrate that there is 5-torsion also in the bottom nonvanishing homology group of the matching complex M14 on 14 vertices. Combining our observation with results due to Bouc and to Shareshian and Wachs, we c...

We study the closed neighborhood ideals and dominating of graphs, in particular, some classes trees cycles. prove that are normally torsion free. The cycles fail to be However, we admit (strong) persistance property nearly