scholarly journals On the spectrum of Cayley graphs

2020 ◽  
Vol 30 (2) ◽  
pp. 194-206
Author(s):  
M. Ghorbani ◽  
◽  
M. Songhori ◽  
Keyword(s):  
Adjacency Matrix ◽  
Abelian Groups ◽  
Cayley Graphs ◽  
Prime Numbers ◽  
Character Table ◽  
Connected Subset

The set of eigenvalues of the adjacency matrix of a graph is called the spectrum of it. In the present paper, we introduce the spectrum of Cayley graphs of order pqr in terms of character table, where p,q,r are prime numbers. We also, stablish some properties of Cayley graphs of non-abelian groups with a normal symmetric connected subset.

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 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 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 An Interesting Property of a Class of Circulant Graphs

2017 ◽  
Vol 2017 ◽  
pp. 1-4
Author(s):  
Seyed Morteza Mirafzal ◽  
Ali Zafari
Keyword(s):  
Adjacency Matrix ◽  
Cayley Graphs ◽  
Additive Group ◽  
Interesting Property ◽  
Circulant Graphs ◽  
Transitive Graph ◽  
Matrix Spectrum ◽  
Adjacency Matrix Spectrum

Suppose thatΠ=Cay(Zn,Ω)andΛ=Cay(Zn,Ψm)are two Cayley graphs on the cyclic additive groupZn, wherenis an even integer,m=n/2+1,Ω=t∈Zn∣t  is  odd, andΨm=Ω∪{n/2}are the inverse-closed subsets ofZn-0. In this paper, it is shown thatΠis a distance-transitive graph, and, by this fact, we determine the adjacency matrix spectrum ofΠ. Finally, we show that ifn≥8andn/2is an even integer, then the adjacency matrix spectrum ofΛisn/2+11,1-n/21,1n-4/2,-1n/2(we write multiplicities as exponents).

scholarly journals Cubic bi-Cayley graphs over abelian groups

2014 ◽  
Vol 36 ◽  
pp. 679-693 ◽  
Author(s):  
Jin-Xin Zhou ◽  
Yan-Quan Feng
Keyword(s):  
Abelian Groups ◽  
Cayley Graphs
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