Long-Scale Ollivier Ricci Curvature of Graphs

D. Cushing; S. Kamtue

doi:10.1515/agms-2019-0003

Long-Scale Ollivier Ricci Curvature of Graphs

Analysis and Geometry in Metric Spaces ◽

10.1515/agms-2019-0003 ◽

2019 ◽

Vol 7 (1) ◽

pp. 22-44

Author(s):

D. Cushing ◽

S. Kamtue

Keyword(s):

Ricci Curvature ◽

Piecewise Linear ◽

Cartesian Product ◽

Regular Graphs ◽

Short Scale

Abstract We study the long-scale Ollivier Ricci curvature of graphs as a function of the chosen idleness. Similarly to the previous work on the short-scale case, we show that this idleness function is concave and piecewise linear with at most 3 linear parts. We provide bounds on the length of the first and last linear pieces. We also study the long-scale curvature for the Cartesian product of two regular graphs.

Download Full-text

Construction of Antimagic Labeling for the Cartesian Product of Regular Graphs

Mathematics in Computer Science ◽

10.1007/s11786-011-0084-3 ◽

2011 ◽

Vol 5 (1) ◽

pp. 81-87 ◽

Cited By ~ 2

Author(s):

Oudone Phanalasy ◽

Mirka Miller ◽

Costas S. Iliopoulos ◽

Solon P. Pissis ◽

Elaheh Vaezpour

Keyword(s):

Cartesian Product ◽

Regular Graphs ◽

Antimagic Labeling

Download Full-text

Bakry–Émery Curvature Functions on Graphs

Canadian Journal of Mathematics ◽

10.4153/cjm-2018-015-4 ◽

2019 ◽

Vol 72 (1) ◽

pp. 89-143 ◽

Cited By ~ 4

Author(s):

David Cushing ◽

Shiping Liu ◽

Norbert Peyerimhoff

Keyword(s):

Lower Bound ◽

Spectral Properties ◽

Cayley Graphs ◽

Cartesian Product ◽

Upper And Lower Bounds ◽

Curvature Function ◽

Regular Graphs ◽

Weighted Graphs ◽

Local Structures ◽

Local Properties

AbstractWe study local properties of the Bakry–Émery curvature function ${\mathcal{K}}_{G,x}:(0,\infty ]\rightarrow \mathbb{R}$ at a vertex $x$ of a graph $G$ systematically. Here ${\mathcal{K}}_{G,x}({\mathcal{N}})$ is defined as the optimal curvature lower bound ${\mathcal{K}}$ in the Bakry–Émery curvature-dimension inequality $CD({\mathcal{K}},{\mathcal{N}})$ that $x$ satisfies. We provide upper and lower bounds for the curvature functions, introduce fundamental concepts like curvature sharpness and $S^{1}$-out regularity, and relate the curvature functions of $G$ with various spectral properties of (weighted) graphs constructed from local structures of $G$. We prove that the curvature functions of the Cartesian product of two graphs $G_{1},G_{2}$ are equal to an abstract product of curvature functions of $G_{1},G_{2}$. We explore the curvature functions of Cayley graphs and many particular (families of) examples. We present various conjectures and construct an infinite increasing family of 6-regular graphs which satisfy $CD(0,\infty )$ but are not Cayley graphs.

Download Full-text

Condensed Ricci curvature of complete and strongly regular graphs

Involve a Journal of Mathematics ◽

10.2140/involve.2020.13.559 ◽

2020 ◽

Vol 13 (4) ◽

pp. 559-576

Author(s):

Vincent Bonini ◽

Conor Carroll ◽

Uyen Dinh ◽

Sydney Dye ◽

Joshua Frederick ◽

...

Keyword(s):

Ricci Curvature ◽

Strongly Regular Graphs ◽

Regular Graphs ◽

Strongly Regular

Download Full-text

Some Methods for constructing infinite Families of Quasi-Strongly regular Graphs

Recent Advances in Computer Science and Communications ◽

10.2174/2666255813999200628093740 ◽

2020 ◽

Vol 13 ◽

Author(s):

Gholam Hassan Shirdel ◽

Adel Asgari

Keyword(s):

Cartesian Product ◽

Necessary Condition ◽

Strongly Regular Graphs ◽

Regular Graphs ◽

Strongly Regular

: In this article, we examined some method of constructing infinite families of semi-strongly regular graphs, Also we obtained a necessary condition for the composition of several graphs to be semi-strongly regular graphs, and using it, we have constructed some infinite families of semi-strongly regular graphs, Also by using the Cartesian product of two graphs, we have constructed some infinite families of semi-strongly regular graphs.

Download Full-text

Super edge- and point-connectivities of the Cartesian product of regular graphs

Networks ◽

10.1002/net.10037 ◽

2002 ◽

Vol 40 (2) ◽

pp. 91-96 ◽

Cited By ~ 18

Author(s):

Bih-Sheue Shieh

Keyword(s):

Cartesian Product ◽

Regular Graphs

Download Full-text

The Number of Spanning Trees of the Cartesian Product of Regular Graphs

Mathematical Problems in Engineering ◽

10.1155/2014/750618 ◽

2014 ◽

Vol 2014 ◽

pp. 1-9

Author(s):

Mei-Hui Wu ◽

Long-Yeu Chung

Keyword(s):

Simple Formula ◽

Spanning Trees ◽

Cartesian Product ◽

Regular Graphs ◽

Electrical Circuits ◽

Important Measure ◽

Computational Perspective ◽

Regular Networks ◽

Number Of Spanning Trees

The number of spanning trees in graphs or in networks is an important issue. The evaluation of this number not only is interesting from a mathematical (computational) perspective but also is an important measure of reliability of a network or designing electrical circuits. In this paper, a simple formula for the number of spanning trees of the Cartesian product of two regular graphs is investigated. Using this formula, the number of spanning trees of the four well-known regular networks can be simply taken into evaluation.

Download Full-text

Perfect state transfer, integral circulants, and join of graphs

Quantum Information and Computation ◽

10.26421/qic10.3-4-10 ◽

2010 ◽

Vol 10 (3&4) ◽

pp. 325-342

Author(s):

R.-J. Angeles-Canul ◽

R.M. Norton ◽

M.C. Opperman ◽

C.C. Paribello ◽

M.C. Russell ◽

...

Keyword(s):

Applied Mathematics ◽

Cartesian Product ◽

Specific Class ◽

Perfect State ◽

Regular Graphs ◽

State Transfer ◽

Join Of Graphs ◽

Perfect State Transfer ◽

Transfer Integral ◽

Product Of Graphs

We propose new families of graphs which exhibit quantum perfect state transfer. Our constructions are based on the join operator on graphs, its circulant generalizations, and the Cartesian product of graphs. We build upon the results of Ba\v{s}i\'{c} and Petkovi\'{c} ({\em Applied Mathematics Letters} {\bf 22}(10):1609-1615, 2009) and construct new integral circulants and regular graphs with perfect state transfer. More specifically, we show that the integral circulant $\textsc{ICG}_{n}(\{2,n/2^{b}\} \cup Q)$ has perfect state transfer, where $b \in \{1,2\}$, $n$ is a multiple of $16$ and $Q$ is a subset of the odd divisors of $n$. Using the standard join of graphs, we also show a family of double-cone graphs which are non-periodic but exhibit perfect state transfer. This class of graphs is constructed by simply taking the join of the empty two-vertex graph with a specific class of regular graphs. This answers a question posed by Godsil (arxiv.org math/08062074).

Download Full-text