scholarly journals Directed Strongly Regular Cayley Graphs over Metacyclic Groups of Order 4n

Mathematics ◽  
2019 ◽  
Vol 7 (11) ◽  
pp. 1011 ◽  
Author(s):  
Tao Cheng ◽  
Lihua Feng ◽  
Weijun Liu
Keyword(s):  
Positive Integer ◽  
Cayley Graphs ◽  
Metacyclic Group ◽  
Metacyclic Groups ◽  
Strongly Regular

We construct several new families of directed strongly regular Cayley graphs (DSRCGs) over the metacyclic group M 4 n = ⟨ a , b | a n = b 4 = 1 , b − 1 a b = a − 1 ⟩ , some of which generalize those earlier constructions. For a prime p and a positive integer α > 1 , for some cases, we characterize the DSRCGs over M 4 p α .

scholarly journals Locally Primitive Normal Cayley Graphs of Metacyclic Groups

10.37236/185 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Jiangmin Pan
Keyword(s):  
Dihedral Group ◽  
Cayley Graphs ◽  
Complete Characterization ◽  
Metacyclic Group ◽  
Metacyclic Groups ◽  
Normal Cayley Graphs

A complete characterization of locally primitive normal Cayley graphs of metacyclic groups is given. Namely, let $\Gamma={\rm Cay}(G,S)$ be such a graph, where $G\cong{\Bbb Z}_m.{\Bbb Z}_n$ is a metacyclic group and $m=p_1^{r_1}p_2^{r_2}\cdots p_t^{r_t}$ such that $p_1 < p_2 < \dots < p_t$. It is proved that $G\cong D_{2m}$ is a dihedral group, and $val(\Gamma)=p$ is a prime such that $p|(p_1(p_1-1),p_2-1,\dots,p_t-1)$. Moreover, three types of graphs are constructed which exactly form the class of locally primitive normal Cayley graphs of metacyclic groups.

scholarly journals Extension of Strongly Regular Graphs

10.37236/878 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Ralucca Gera ◽  
Jian Shen
Keyword(s):  
Positive Integer ◽  
Nonnegative Integer ◽  
Regularity Property ◽  
Strongly Regular Graphs ◽  
Regular Graphs ◽  
Strongly Regular ◽  
Vertex Degrees

The Friendship Theorem states that if any two people in a party have exactly one common friend, then there exists a politician who is a friend of everybody. In this paper, we generalize the Friendship Theorem. Let $\lambda$ be any nonnegative integer and $\mu$ be any positive integer. Suppose each pair of friends have exactly $\lambda$ common friends and each pair of strangers have exactly $\mu$ common friends in a party. The corresponding graph is a generalization of strongly regular graphs obtained by relaxing the regularity property on vertex degrees. We prove that either everyone has exactly the same number of friends or there exists a politician who is a friend of everybody. As an immediate consequence, this implies a recent conjecture by Limaye et. al.

scholarly journals Directed strongly regular Cayley graphs on dihedral groups

2021 ◽  
Vol 391 ◽  
pp. 125651
Author(s):  
Yiqin He ◽  
Bicheng Zhang ◽  
Rongquan Feng
Keyword(s):  
Cayley Graphs ◽  
Dihedral Groups ◽  
Strongly Regular

On semireversible rings

2019 ◽  
pp. 2150018
Author(s):  
Debraj Roy ◽  
Tikaram Subedi
Keyword(s):  
Positive Integer ◽  
Matrix Rings ◽  
New Class ◽  
Ring Of Fractions ◽  
Strongly Regular

In this paper, we introduce and study a new class of rings which we call semireversible rings. A ring [Formula: see text] is called semireversible if for any [Formula: see text] implies there exists a positive integer [Formula: see text] such that [Formula: see text]. We observe that the class of semireversible rings strictly lies between the class of central reversible rings and weakly reversible rings. Some relations are provided between semireversible rings and many other known classes of rings. Some extensions of semireversible rings such as ring of fractions, Dorroh extension, subrings of matrix rings are investigated. Finally, we study semireversible rings via modules over them wherein among other results, we prove that a semireversible left (right) SF-ring is strongly regular.

scholarly journals Strongly regular Cayley graphs with λ − μ = −1

1994 ◽  
Vol 67 (1) ◽  
pp. 116-125 ◽  
Author(s):  
K.T Arasu ◽  
D Jungnickel ◽  
S.L Ma ◽  
A Pott
Keyword(s):  
Cayley Graphs ◽  
Strongly Regular

scholarly journals Automorphisms of Cayley graphs of metacyclic groups of prime-power order

2001 ◽  
Vol 71 (2) ◽  
pp. 223-232 ◽  
Author(s):  
Caiheng Li ◽  
Hyo-Seob Sim
Keyword(s):  
Graph Theory ◽  
Cayley Graph ◽  
Prime Power ◽  
Automorphism Groups ◽  
Cayley Graphs ◽  
Prime Power Order ◽  
Subsequent Work ◽  
Metacyclic Groups

AbstractThis paper inverstigates the automorphism groups of Cayley graphs of metracyclicp-gorups. A characterization is given of the automorphism groups of Cayley grahs of a metacyclicp-group for odd primep. In particular, a complete determiniation of the automophism group of a connected Cayley graph with valency less than 2pof a nonabelian metacyclicp-group is obtained as a consequence. In subsequent work, the result of this paper has been applied to solve several problems in graph theory.

scholarly journals On isomorphisms of connected Cayley graphs, III

1998 ◽  
Vol 58 (1) ◽  
pp. 137-145 ◽  
Author(s):  
Cai Heng Li
Keyword(s):  
Finite Group ◽  
Positive Integer ◽  
Simple Group ◽  
Prime Divisor ◽  
Cayley Graphs ◽  
Simple Groups ◽  
Linear Groups ◽  
Finite Simple Groups ◽  
Natural Question ◽  
A Finite Group

For a finite group G and a subset S of G which does not contain the identity of G, we use Cay(G, S) to denote the Cayley graph of G with respect to S. For a positive integer m, the group G is called a (connected) m-DCI-group if for any (connected) Cayley graphs Cay(G, S) and Cay(G, T) of out-valency at most m, Sσ = T for some σ ∈ Aut(G) whenever Cay(G, S) ≅ Cay(G, T). Let p(G) be the smallest prime divisor of |G|. It was previously shown that each finite group G is a connected m-DCI-group for m ≤ p(G) − 1 but this is not necessarily true for m = p(G). This leads to a natural question: which groups G are connected p(G)-DCI-groups? Here we conjecture that the answer of this question is positive for finite simple groups, that is, finite simple groups are all connected 2-DCI-groups. We verify this conjecture for the linear groups PSL(2, q). Then we prove that a nonabelian simple group G is a 2-DCI-group if and only if G = A5.

Automorphisms of a Family of Cubic Graphs

2013 ◽  
Vol 20 (03) ◽  
pp. 495-506 ◽  
Author(s):  
Jin-Xin Zhou ◽  
Mohsen Ghasemi
Keyword(s):  
Automorphism Group ◽  
Positive Integer ◽  
Cayley Graph ◽  
Regular Representation ◽  
Cayley Graphs ◽  
Cubic Graphs ◽  
Full Automorphism Group ◽  
Vertex Set ◽  
The Right ◽  
Normal Cayley Graphs

A Cayley graph Cay (G,S) on a group G with respect to a Cayley subset S is said to be normal if the right regular representation R(G) of G is normal in the full automorphism group of Cay (G,S). For a positive integer n, let Γn be a graph with vertex set {xi,yi|i ∈ ℤ2n} and edge set {{xi,xi+1}, {yi,yi+1}, {x2i,y2i+1}, {y2i,x2i+1}|i ∈ ℤ2n}. In this paper, it is shown that Γn is a Cayley graph and its full automorphism group is isomorphic to [Formula: see text] for n=2, and to [Formula: see text] for n > 2. Furthermore, we determine all pairs of G and S such that Γn= Cay (G,S) is non-normal for G. Using this, all connected cubic non-normal Cayley graphs of order 8p are constructed explicitly for each prime p.

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