In this paper, inspired by the work of Tan et al. (2010) 36, Chee et al. (2011) 11 and Hyun et al. (2020) 19, we propose two new constructions of strongly regular graphs on finite fields by using ...weakly regular bent functions, which generalize the results in the existing references. We obtain two families of strongly regular graphs with flexible parameters. We are also able to obtain a 3-class amorphic association scheme. Moreover, we show that many of the strongly regular graphs which we construct are Ramanujan graphs.
Perfect state transfer on graphs has attracted extensive attention due to its application in quantum information and quantum computation. A graph is a semi-Cayley graph over a group
G
if it admits
G
...as a semiregular subgroup of the full automorphism group with two orbits of equal size. A semi-Cayley graph
SC
(
G
,
R
,
L
,
S
) is called quasi-abelian if each of
R
,
L
and
S
is a union of some conjugacy classes of
G
. This paper establishes necessary and sufficient conditions for a quasi-abelian semi-Cayley graph to have perfect state transfer. As a corollary, it is shown that if a quasi-abelian semi-Cayley graph over a finite group
G
has perfect state transfer between distinct vertices
g
and
h
, and
G
has a faithful irreducible character, then
g
h
-
1
lies in the center of
G
and
g
h
=
h
g
; in particular,
G
cannot be a non-abelian simple group. We also characterize quasi-abelian Cayley graphs over arbitrary groups having perfect state transfer, which is a generalization of previous works on Cayley graphs over abelian groups, dihedral groups, semi-dihedral groups and dicyclic groups.
If all the eigenvalues of the Hermitian-adjacency matrix of a mixed graph are integers, then the mixed graph is called H-integral. If all the eigenvalues of the (0,1)-adjacency matrix of a mixed ...graph are Gaussian integers, then the mixed graph is called Gaussian integral. For any finite group Γ, we characterize the set S for which the normal mixed Cayley graph Cay(Γ,S) is H-integral. We further prove that a normal mixed Cayley graph is H-integral if and only if the mixed graph is Gaussian integral.
We generalise the standard constructions of a Cayley graph in terms of a group presentation by allowing some vertices to obey different relators than others. The resulting notion of presentation ...allows us to represent every vertex-transitive graph.
How to classify all the integral graphs is a challenging work as suggested by Harary and Schwenk. In this paper, we focus on one non-abelian group—the dicyclic group T4n=〈a,b|a2n=1,an=b2,b−1ab=a−1〉, ...and consider its corresponding Cayley graphs. With the help of its character table, we first obtain a necessary and sufficient condition for the integrality of Cayley graphs over T4n. Then we obtain several simple sufficient conditions for the integrality of Cayley graphs over T4n in terms of the Boolean algebra of 〈a〉. As a byproduct, we determine a few infinite families of connected integral Cayley graphs over T4n. At last, for a prime p, we completely determine all integral Cayley graphs over the dicyclic group T4p.
A Pℓ-decomposition of a graph G is a set of paths with ℓ edges in G that cover the edge set of G. Favaron, Genest, and Kouider (2010) conjectured that every (2k+1)-regular graph that contains a ...perfect matching admits a P2k+1-decomposition. They also verified this conjecture for 5-regular graphs without cycles of length 4. In 2015, Botler, Mota, and Wakabayashi verified this conjecture for 5-regular graphs without triangles. In this paper, we verify it for (2k+1)-regular graphs that contain the kth power of a spanning cycle; and for 5-regular graphs that contain special spanning 4-regular Cayley graphs.
Classifying integral graphs is a hard problem that initiated by Harary and Schwenk in 1974. In this paper, with the help of character table, we treat the corresponding problem for Cayley graphs over ...the semi-dihedral group SD8n = 〈a, b | a4n = b² = 1, bab = a
2n−1⟩, n ≥ 2. We present several necessary and sufficient conditions for the integrality of Cayley graphs over SD8n, we also obtain some simple sufficient conditions for the integrality of Cayley graphs over SD8n in terms of the Boolean algebra of 〈a〉. In particular, we give the sufficient conditions for the integrality of Cayley graphs over semi-dihedral groups
SD
2
n
(
n
≥
4
)
and SD8p for a prime p, from which we determine several infinite classes of integral Cayley graphs over
SD
2
n
and SD8p.
Absorption cayley graph Deepa Sinha; Deepakshi Sharma
Electronic notes in discrete mathematics,
September 2016, 2016-09-00, Volume:
53
Journal Article
Let R be a commutative ring with two binary operators addition (+) and multiplication (.). Then Zn is a ring of integers modulo n, where n is a positive integer. A Absorption Cayley graph denoted by ...Ω(Zn) is a graph whose vertex set is Zn, the integer modulo n and edge set E={ab:a+b∈S}, where S={a∈Zn:ab=ba=afor anyb∈Zn,b≠a,b≠1}. Here ab=a is the absorption property as b is absorbed in a. We study the characterization of Absorption cayley graphs along with its properties such as connectedness, degree, diameter, planarity, girth, regularity.
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