scholarly journals Improved Expansion of Random Cayley Graphs

2004 ◽  
Vol Vol. 6 no. 2 ◽  
Author(s):  
Po-Shen Loh ◽  
Leonard J. Schulman
Keyword(s):  
Cayley Graph ◽  
Adjacency Matrix ◽  
Abelian Groups ◽  
Cayley Graphs ◽  
Expected Value ◽  
Small Eigenvalue ◽  
Irreducible Representations ◽  
Absolute Value ◽  
Graph Expansion ◽  
International Audience

International audience In Random Cayley Graphs and Expanders, N. Alon and Y. Roichman proved that for every ε > 0 there is a finite c(ε ) such that for any sufficiently large group G, the expected value of the second largest (in absolute value) eigenvalue of the normalized adjacency matrix of the Cayley graph with respect to c(ε ) log |G| random elements is less than ε . We reduce the number of elements to c(ε )log D(G) (for the same c), where D(G) is the sum of the dimensions of the irreducible representations of G. In sufficiently non-abelian families of groups (as measured by these dimensions), log D(G) is asymptotically (1/2)log|G|. As is well known, a small eigenvalue implies large graph expansion (and conversely); see Tanner84 and AlonMilman84-2,AlonMilman84-1. For any specified eigenvalue or expansion, therefore, random Cayley graphs (of sufficiently non-abelian groups) require only half as many edges as was previously known.

scholarly journals On the Characteristic Polynomial of $n$-Cayley Digraphs

10.37236/3105 ◽  
2013 ◽  
Vol 20 (3) ◽  
Author(s):  
Majid Arezoomand ◽  
Bijan Taeri
Keyword(s):  
Cayley Graph ◽  
Characteristic Polynomial ◽  
Adjacency Matrix ◽  
Cayley Graphs ◽  
Irreducible Representations ◽  
Cayley Digraph ◽  
Semiregular Subgroup

A digraph $\Gamma$ is called $n$-Cayley digraph over a group $G$‎, ‎if there exists a semiregular subgroup $R_G$ of Aut$(\Gamma)$ isomorphic to $G$ with $n$ orbits‎. ‎In this paper‎, ‎we represent the adjacency matrix of $\Gamma$ as a diagonal block‎ ‎matrix in terms of irreducible representations of $G$ and determine its characteristic polynomial‎. ‎As corollaries of this result we find‎:  ‎the spectrum of  semi-Cayley graphs over abelian groups‎, ‎a relation between the characteristic polynomial of an $n$-Cayley graph and its complement‎, ‎and   the spectrum of‎ ‎Calye graphs over groups with cyclic subgroups‎. ‎Finally we determine the eigenspace of $n$-Cayley digraphs and their main eigenvalues‎.

scholarly journals Integral Mixed Cayley Graphs over Abelian Groups

2021 ◽  
Vol 28 (4) ◽  
Author(s):  
Monu Kadyan ◽  
Bikash Bhattacharjya
Keyword(s):  
Abelian Group ◽  
Cayley Graph ◽  
Adjacency Matrix ◽  
Abelian Groups ◽  
Cayley Graphs ◽  
Mixed Graph ◽  
Vertex Set ◽  
Gamma S

A mixed graph is said to be integral if all the eigenvalues of its Hermitian adjacency matrix are integer. Let $\Gamma$ be an abelian group. The mixed Cayley graph $Cay(\Gamma,S)$ is a mixed graph on the vertex set $\Gamma$ and edge set $\left\{ (a,b): b-a\in S \right\}$, where $0\not\in S$. We characterize integral mixed Cayley graph $Cay(\Gamma,S)$ over an abelian group $\Gamma$ in terms of its connection set $S$.

Some properties of subgroup complementary addition Cayley graphs on abelian groups

2021 ◽  
Author(s):  
Naveen Palanivel ◽  
Chithra A. Velu
Keyword(s):  
Cayley Graph ◽  
Abelian Groups ◽  
Cayley Graphs ◽  
Graph Invariants

In this paper, we introduce subgroup complementary addition Cayley graph [Formula: see text] and compute its graph invariants. Also, we prove that [Formula: see text] if and only if [Formula: see text] for all [Formula: see text] where [Formula: see text].

scholarly journals Isomorphisms of Cayley digraphs of Abelian groups

1998 ◽  
Vol 57 (2) ◽  
pp. 181-188 ◽  
Author(s):  
Cai Heng Li
Keyword(s):  
Finite Group ◽  
Positive Integer ◽  
Cayley Graph ◽  
Prime Divisor ◽  
Open Problem ◽  
Abelian Groups ◽  
Cayley Graphs ◽  
Vertex Set ◽  
A Finite Group

For a finite group G and a subset S of G with 1 ∉ S, the Cayley graph Cay(G, S) is the digraph with vertex set G such that (x, y) is an arc if and only if yx−1 ∈ S. The Cayley graph Cay(G, S) is called a CI-graph if, for any T ⊂ G, whenever Cay (G, S) ≅ Cay(G, T) there is an element a σ ∈ Aut(G) such that Sσ = T. For a positive integer m, G is called an m-DCI-group if all Cayley graphs of G of valency at most m are CI-graphs; G is called a connected m-DCI-group if all connected Cayley graphs of G of valency at most m are CI-graphs. The problem of determining Abelian m-DCI-groups is a long-standing open problem. It is known from previous work that all Abelian m-DCI-groups lie in an explicitly determined class of Abelian groups. First we reduce the problem of determining Abelian m-DCI-groups to the problem of determining whether every subgroup of a member of is a connected m-DCI-group. Then (for a finite group G, letting p be the least prime divisor of |G|,) we completely classify Abelian connected (p + 1)-DCI-groups G, and as a corollary, we completely classify Abelian m-DCI-groups G for m ≤ p + 1. This gives many earlier results when p = 2.

scholarly journals Diameters of random Cayley graphs of finite nilpotent groups

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Daniel El-Baz ◽  
Carlo Pagano
Keyword(s):  
Nilpotent Group ◽  
Cayley Graph ◽  
Abelian Groups ◽  
Cayley Graphs ◽  
Nilpotent Groups ◽  
Nilpotency Class ◽  
Limiting Distribution ◽  
General Inequality ◽  
Bounded Rank

Abstract We prove the existence of a limiting distribution for the appropriately rescaled diameters of random undirected Cayley graphs of finite nilpotent groups of bounded rank and nilpotency class, thus extending a result of Shapira and Zuck which dealt with the case of abelian groups. The limiting distribution is defined on a space of unimodular lattices, as in the case of random Cayley graphs of abelian groups. Our result, when specialised to a certain family of unitriangular groups, establishes a very recent conjecture of Hermon and Thomas. We derive this as a consequence of a general inequality, showing that the diameter of a Cayley graph of a nilpotent group is governed by the diameter of its abelianisation.

scholarly journals A Classification of Ramanujan Unitary Cayley Graphs

10.37236/478 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Andrew Droll
Keyword(s):  
Cayley Graph ◽  
Regular Graph ◽  
Adjacency Matrix ◽  
Complete Characterization ◽  
Absolute Value ◽  
Ramanujan Graph ◽  
Vertex Set ◽  
Unitary Cayley Graph

The unitary Cayley graph on $n$ vertices, $X_n$, has vertex set ${\Bbb Z}/{n\Bbb Z}$, and two vertices $a$ and $b$ are connected by an edge if and only if they differ by a multiplicative unit modulo $n$, i.e. ${\rm gcd}(a-b,n) = 1$. A $k$-regular graph $X$ is Ramanujan if and only if $\lambda(X) \leq 2\sqrt{k-1}$ where $\lambda(X)$ is the second largest absolute value of the eigenvalues of the adjacency matrix of $X$. We obtain a complete characterization of the cases in which the unitary Cayley graph $X_n$ is a Ramanujan graph.

scholarly journals Distance Powers and Distance Matrices of Integral Cayley Graphs over Abelian Groups

10.37236/2369 ◽  
2012 ◽  
Vol 19 (4) ◽  
Author(s):  
Walter Klotz ◽  
Torsten Sander
Keyword(s):  
Abelian Group ◽  
Cayley Graph ◽  
Abelian Groups ◽  
Cayley Graphs ◽  
Integral Spectrum ◽  
Distance Matrices

It is shown that distance powers of an integral Cayley graph over an abelian group $\Gamma$ are again integral Cayley graphs over $\Gamma$. Moreover, it is proved that distance matrices of integral Cayley graphs over abelian groups have integral spectrum.

scholarly journals Normal edge-transitive and \frac{1}{2}-arc-transitive cayley graphs on non-abelian groups of odd order 3pq, p and q are primes

2018 ◽  
Vol 49 (3) ◽  
pp. 183-194
Author(s):  
Ali Reza Ashrafi ◽  
Bijan Soleimani
Keyword(s):  
Cayley Graph ◽  
Abelian Groups ◽  
Cayley Graphs ◽  
Prime Numbers ◽  
Graph Of Groups ◽  
Odd Order ◽  
Edge Transitive

Suppose $p$ and $q$ are odd prime numbers. In this paper, the connected Cayley graph of groups of order $3pq$, for primes $p$ and $q$, are investigated and all connected normal $\frac{1}{2}-$arc-transitive Cayley graphs of group of these orders will be classified.

scholarly journals Random Cayley Graphs are Expanders: a Simple Proof of the Alon–Roichman Theorem

10.37236/1815 ◽  
2004 ◽  
Vol 11 (1) ◽  
Author(s):  
Zeph Landau ◽  
Alexander Russell
Keyword(s):  
Finite Group ◽  
Cayley Graph ◽  
Group Algebra ◽  
Simple Proof ◽  
Cayley Graphs ◽  
Random Variables ◽  
Irreducible Representations ◽  
Original Proof ◽  
A Finite Group ◽  
Second Eigenvalue

We give a simple proof of the Alon–Roichman theorem, which asserts that the Cayley graph obtained by selecting $c_\varepsilon \log |G|$ elements, independently and uniformly at random, from a finite group $G$ has expected second eigenvalue no more than $\varepsilon$; here $c_\varepsilon$ is a constant that depends only on $\varepsilon$. In particular, such a graph is an expander with constant probability. Our new proof has three advantages over the original proof: (i.) it is extremely simple, relying only on the decomposition of the group algebra and tail bounds for operator-valued random variables, (ii.) it shows that the $\log |G|$ term may be replaced with $\log D$, where $D \leq |G|$ is the sum of the dimensions of the irreducible representations of $G$, and (iii.) it establishes the result above with a smaller constant $c_\varepsilon$.

scholarly journals Which Haar graphs are Cayley graphs?

10.37236/5240 ◽  
2016 ◽  
Vol 23 (3) ◽  
Author(s):  
István Estélyi ◽  
Tomaž Pisanski
Keyword(s):  
Finite Group ◽  
Abelian Group ◽  
Cayley Graph ◽  
Regular Graph ◽  
Abelian Groups ◽  
Cayley Graphs ◽  
Equivalent Condition ◽  
Dihedral Groups ◽  
A Finite Group

For a finite group $G$ and subset $S$ of $G,$ the Haar graph $H(G,S)$ is a bipartite regular graph, defined as a regular $G$-cover of a dipole with $|S|$ parallel arcs labelled by elements of $S$. If $G$ is an abelian group, then $H(G,S)$ is well-known to be a Cayley graph; however, there are examples of non-abelian groups $G$ and subsets $S$ when this is not the case. In this paper we address the problem of classifying finite non-abelian groups $G$ with the property that every Haar graph $H(G,S)$ is a Cayley graph. An equivalent condition for $H(G,S)$ to be a Cayley graph of a group containing $G$ is derived in terms of $G, S$ and $\mathrm{Aut } G$. It is also shown that the dihedral groups, which are solutions to the above problem, are $\mathbb{Z}_2^2,D_3,D_4$ and $D_{5}$. 

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