Graphs Defined on Rings: A Review Madhumitha, S.; Naduvath, Sudev
Mathematics (Basel),
09/2023, Volume:
11, Issue:
17
Journal Article
Peer reviewed
Open access
The study on graphs emerging from different algebraic structures such as groups, rings, fields, vector spaces, etc. is a prominent area of research in mathematics, as algebra and graph theory are two ...mathematical fields that focus on creating and analysing structures. There are numerous studies linking algebraic structures and graphs, which began with the introduction of Cayley graphs of groups. Several algebraic graphs have been defined on rings, a fast-growing area in the literature. In this article, we systematically review the literature on some variants of Cayley graphs that are defined on rings and highlight the properties and characteristics of such graphs, to showcase the research in this area.
Recently, several works by a number of authors have provided characterizations of integral undirected Cayley graphs over generalized dihedral groups and generalized dicyclic groups. We generalize and ...unify these results in two different ways. Firstly, we work over arbitrary non-abelian finite groups admitting an abelian subgroup of index 2. Secondly, our main result actually characterizes integral mixed Cayley graphs over such finite groups, in the spirit of a very recent result of Kadyan–Bhattarcharjya in the abelian case.
We prove an upper bound on the number of pairwise strongly cospectral vertices in a normal Cayley graph, in terms of the multiplicities of its eigenvalues. We use this to determine an explicit bound ...in Cayley graphs of Z2d and Z4d. We also provide some infinite families of Cayley graphs of Z2d with a set of four pairwise strongly cospectral vertices and show that such graphs exist in every dimension.
GCI-groups in the alternating groups Yang, Xu; Liu, Weijun; Chen, Jing ...
Applied mathematics and computation,
06/2017, Volume:
303
Journal Article
Peer reviewed
The concept of generalized Cayley graphs was first introduced by Marušič et al. (1992). In this paper, we will study the isomorphism problem of generalized Cayley graphs. Similar to the concept of ...CI-groups corresponding to Cayley graphs, we define the so-called GCI-groups corresponding to generalized Cayley graphs. The main result we show is that the alternating group An is a GCI-group if and only if n=4.
Fire retainment on Cayley graphs Amir, Gideon; Baldasso, Rangel; Gerasimova, Maria ...
Discrete mathematics,
January 2023, 2023-01-00, Volume:
346, Issue:
1
Journal Article
Peer reviewed
Open access
We study the fire-retaining problem on groups, a quasi-isometry invariant11See Section 2 for some caveats. introduced by Martínez-Pedroza and Prytuła 8, related to the firefighter problem. We prove ...that any Cayley graph with degree-d polynomial growth does not satisfy {f(n)}-retainment, for any f(n)=o(nd−2), matching the upper bound given for the firefighter problem for these graphs. In the exponential growth regime we prove general lower bounds for direct products and wreath products. These bounds are tight, and show that for exponential-growth groups a wide variety of behaviors is possible. In particular, we construct, for any d≥1, groups that satisfy {nd}-retainment but not o(nd)-retainment, as well as groups that do not satisfy sub-exponential retainment.
In this paper, we have obtained the total chromatic number for some classes of Cayley graphs, particularly the Unitary Cayley graphs on even order and some other Circulant graphs. We have also proved ...the Total Coloring Conjecture for some perfect Cayley graphs.
Consider a graph whose vertices are populated by identical objects, together with an algorithm for the time-evolution of the number of objects placed at each of the vertices. The discrete dynamics of ...these objects can be observed and studied using simple and inexpensive laboratory settings. There are many similarities but also many differences between such population dynamics and the quantum dynamics of a quantum particle hopping on the same graph. In this work, we show that a specific decoration of the original graph enables an exact mapping between models of population and quantum dynamics. As such, population dynamics over graphs is yet another classical platform that can simulate quantum effects. Several examples are used to demonstrate this claim.
•Population dynamics can be set by simple deterministic algorithms and observed in inexpensive tabletop experiments.•The generators of such dynamics resemble those of quantum mechanics, with the major difference that the latter involve complex coefficients while the former only allow for positive real coefficients.•We use a certain decoration of the graph to coherently encode complex numbers into four positive numbers and link a quantum dynamics to a population dynamics.•We demonstrate with examples how certain aspects of quantum dynamics can be reproduced with population dynamics.
In this article, we describe the Cayley graphs of pseudo-unitary homogeneous semigroups with zero 0, which are 0-disjoint unions of their subsets indexed by a groupoid.
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