本文通过将彼得森图复制n次并按n阶循环置换连接顶点，构造出无限族6正则图{G_n}，证明G_3和G_4为拉马努金图，并将前五项提交至House of Graphs数据库。
摘要：
我们通过将彼得森图复制n次并按n阶循环置换连接相应顶点，构造出无限族{G_n}（n≥3）的6正则图。每个G_n有10n个顶点、30n条边，其自同构群为阶数为10n的D_{5n}，作用于两个大小为5n的顶点轨道上。这些图具有girth 4和直径⌊n/2⌋+2。我们证明G_3和G_4为拉马努金图，满足|λ_2| ≤ 2√5。前五项（n=3,…,7）已提交至House of Graphs数据库作为条目56324–56328。此构造提供了高度对称的正则图新例子，并向文献贡献了两个新的拉马努金图。所有计算脚本均在线可用以确保完全可重复性。
英文摘要：
We construct an infinite family of 6-regular graphs $\{G_n\}_{n\ge 3}$ by taking $n$ copies of the Petersen graph and wiring corresponding vertices according to an $n$-cycle permutation. Each $G_n$ has $10n$ vertices, $30n$ edges, and automorphism group $D_{5n}$ of order $10n$, acting with two vertex orbits of size $5n$. The graphs have girth $4$ and diameter $\lfloor n/2\rfloor+2$. We prove that $G_3$ and $G_4$ are Ramanujan graphs, satisfying $|λ_2| \le 2\sqrt{5}$. The first five members ($n=3,\dots,7$) have been deposited in the House of Graphs database as entries 56324--56328. This construction provides new examples of highly symmetric regular graphs and contributes two new Ramanujan graphs to the literature. All computational scripts are available online for full reproducibility.
A graph is a core or unretractive if all its endomorphisms are automorphisms. Well-known examples of cores include the Petersen graph and the graph of the dodecahedron -- both generalized Petersen graphs. We characterize the generalized Petersen graphs that are cores. A simple characterization of endomorphism-transitive generalized Petersen graphs follows. This extends the characterization of vertex-transitive generalized Petersen graphs due to Frucht, Graver, and Watkins and solves a problem of Fan and Xie. Moreover, we study generalized Petersen graphs that are (underlying graphs of) Cayley …
The class of generalized Petersen graphs was introduced by Coxeter in the 1950s. Frucht, Graver and Watkins determined the automorphism groups of generalized Petersen graphs in 1971, and much later, Nedela and Škoviera and (independently) Lovrečič-Saražin characterised those which are Cayley graphs. In this paper we extend the class of generalized Petersen graphs to a class of GI-graphs. For any positive integer n and any sequence j_0,j_1,....,j_{t-1} of integers mod n, the GI-graph GI(n;j_0,j_1,....,j_{t-1}) is a (t+1)-valent graph on the vertex set Z_t x Z_n, with edges of two kinds: - an ed…
In \cite{Chan95}, the authors classified the 2-extendable abelian Cayley graphs and posed the problem of characterizing all 2-extendable Cayley graphs. We first show that a connected bipartite Cayley (vertex-transitive) graph is 2-extendable if and only if it is not a cycle. It is known that a non-bipartite Cayley (vertex-transitive) graph is 2-extendable when it is of minimum degree at least five \cite{sun}. We next classify all 2-extendable cubic non-bipartite Cayley graphs and obtain that: a cubic non-bipartite Cayley graph with girth $g$ is 2-extendable if and only if $g\geq 4$ and it does…
The family of generalized Petersen graphs $G(n, k)$, introduced by Coxeter et al. [4] and named by Mark Watkins (1969), is a family of cubic graphs formed by connecting the vertices of a regular polygon to the corresponding vertices of a star polygon. The Kronecker cover $\mathrm{KC}(G)$ of a simple undirected graph $G$ is a a special type of bipartite covering graph of $G$, isomorphic to the direct (tensor) product of $G$ and $K_2$. We characterize all the members of generalized Petersen graphs that are Kronecker covers, and describe the structure of their respective quotients. We observe tha…
In this paper, we study self-complementary distance-regular Cayley graphs over abelian groups. We prove that if a regular graph is self-complementary distance-regular, then it is self-complementary strongly regular. We also deal with self-complementary strongly regular Cayley graphs over abelian groups and give an example of a self-complementary strongly regular Cayley graph over a non-elementary abelian group.
The characterization of distance-regular Cayley graphs originated from the problem of identifying strongly regular Cayley graphs, or equivalently, regular partial difference sets. In this paper, a classification of distance-regular Cayley graphs on dicyclic groups is obtained. More specifically, it is shown that every distance-regular Cayley graph on a dicyclic group is a complete graph, a complete multipartite graph, or a non-antipodal bipartite distance-regular graph with diameter $3$ satisfying some additional conditions.
We consider the problem of which distance-regular graphs with small valency are Cayley graphs. We determine the distance-regular Cayley graphs with valency at most $4$, the Cayley graphs among the distance-regular graphs with known putative intersection arrays for valency $5$, and the Cayley graphs among all distance-regular graphs with girth $3$ and valency $6$ or $7$. We obtain that the incidence graphs of Desarguesian affine planes minus a parallel class of lines are Cayley graphs. We show that the incidence graphs of the known generalized hexagons are not Cayley graphs, and neither are som…
Recently, regular Cayley maps of cyclic groups and dihedral groups have been classified. A nature question is to classify regular Cayley maps of elementary abelian $p$-groups $Z_p^n$. In this paper, a complete classification of regular Cayley maps of $Z_p^n$ is given and moreover, the number of these maps and their genera are enumerated.
In this paper, we give a construction of strongly regular Cayley graphs on the additive groups of finite fields based on three-valued Gauss periods. As consequences, we obtain two infinite families and one sporadic example of new strongly regular Cayley graphs. This construction can be viewed as a generalization of that of strongly regular Cayley graphs obtained in \cite{BLMX}.
A Cayley map is a 2-cell embedding of a Cayley graph into an orientable surface with the same local orientation induced by a cyclic permutation of generators at each vertex. In this paper, we provide classifications of prime-valent regular Cayley maps on abelian groups, dihedral groups and dicyclic groups. Consequently, we show that all prime-valent regular Cayley maps on dihedral groups are balanced and all prime-valent regular Cayley maps on abelian groups are either balanced or anti-balanced. Furthermore, we prove that there is no prime-valent regular Cayley map on any dicyclic group.
In this paper, we give a new lifting construction of "hyperbolic" type of strongly regular Cayley graphs. Also we give new constructions of strongly regular Cayley graphs over the additive groups of finite fields based on partitions of subdifference sets of the Singer difference sets. Our results unify some recent constructions of strongly regular Cayley graphs related to $m$-ovoids and $i$-tight sets in finite geometry. Furthermore, some of the strongly regular Cayley graphs obtained in this paper are new or nonisomorphic to known strongly regular graphs with the same parameters.
In this paper,we construct some directed strongly regular Cayley graphs on dihedral groups,these generalizes some earlier constructions.We also characterize some certain directed strongly regular Cayley graphs on dihedral groups $D_{p^α}$,where $p$ is a prime and $α\geqslant 1$ is a positive integer.
In this paper, we characterize some certain directed strongly regular Cayley graphs on Dihedral groups $D_{n}$, where $n\geqslant 3$ is a positive integer.
In this paper, generalizing the result in \cite{GXY}, we construct strongly regular Cayley graphs by using union of cyclotomic classes of $\F_q$ and Gauss sums of index $w$, where $w\geq 2$ is even. In particular, we obtain three infinite families of strongly regular graphs with new parameters.
We give a construction of strongly regular Cayley graphs on finite fields $\F_q$ by using union of cyclotomic classes and index 4 Gauss sums. In particular, we obtain two infinite families of strongly regular graphs with new parameters.
We study bipartite distance-regular Cayley graphs with diameter three or four. We give sufficient conditions under which a bipartite Cayley graph can be constructed on the semidirect product of a group -- the part of this bipartite Cayley graph which contains the identity element -- and $\mathbb{Z}_{2}$. We apply this to the case of bipartite distance-regular Cayley graphs with diameter three, and consider cases where the sufficient conditions are not satisfied for some specific groups such as the dihedral group. We also extend a result by Miklavič and Potočnik that relates difference sets to …
Directed strongly regular graphs were introduced by Duval in 1998 as one of the possible generalization of classical strongly regular graphs to the directed case. Duval also provided several construction methods for directed strongly regular graphs. In this paper, an infinite family of directed strongly regular graphs is constructed, as generalized Cayley graphs of cyclic groups.
A signed graph $Σ=(G,σ)$ is said to be parity signed if there exists a bijection $f : V(G) \rightarrow \{1,2,...,|V(G)|\}$ such that $σ(uv)=+$ if and only if $f(u)$ and $f(v)$ are of same parity, where $uv$ is an edge of $G$. The rna number of a graph $G$, denoted $σ^{-}(G)$, is the minimum number of negative edges among all possible parity signed graphs over $G$. The rna number is also equal to the minimum cut size that has nearly equal sides. In this paper, for generalized Petersen graph $P(n,k)$, we prove that $3 \leq σ^{-}(P(n,k)) \leq n$ and these bounds are sharp. The exact value of $σ^{…
In 1988, Duval introduced the concept of directed strongly regular graphs, which can be viewed as a directed graph version of strongly regular graphs. Such directed graphs have similar structural and algebraic properties to strongly regular graphs. In the past three decades, it was found that Cayley graphs, especially those over dihedral groups, play a key role in the construction of directed strongly regular graphs. In this paper, we focus on the characterization of directed strongly regular Cayley graphs over more general groups. Let $G$ be a non-abelian group with an abelian subgroup of ind…
We give two "lifting" constructions of strongly regular Cayley graphs. In the first construction we "lift" a cyclotomic strongly regular graph by using a subdifference set of the Singer difference set. The second construction uses quadratic forms over finite fields and it is a common generalization of the construction of the affine polar graphs \cite{CK86} and a construction of strongly regular Cayley graphs given in \cite{FWXY}. The two constructions are related in the following way: The second construction can be viewed as a recursive construction, and the strongly regular Cayley graphs obta…
In this paper, we show that every bipartite distance-regular Cayley graph with diameter $3$ can be constructed on the semidirect product of a group and $\mathbb{Z}_{2}$, except possibly for one case.
We classify the distance-regular Cayley graphs with least eigenvalue $-2$ and diameter at most three. Besides sporadic examples, these comprise of the lattice graphs, certain triangular graphs, and line graphs of incidence graphs of certain projective planes. In addition, we classify the possible connection sets for the lattice graphs and obtain some results on the structure of distance-regular Cayley line graphs of incidence graphs of generalized polygons.
Strongly regular Cayley graphs with Paley parameters over abelian groups of rank 2 were studied in [J.A Davis, Partial difference sets in p-groups, Arch.Math.63 (1994) 103-110; K.H Leung, S.L. Ma, Partial difference sets with Paley parameters, Bull. London Math Soc. 27 (1995) 553-564]. It was shown that such graphs exist iff the corresponding group is isomorphic to ${\mathbb Z}_{p^n} \oplus {\mathbb Z}_{p^n}$, where $p$ is an odd prime. In this paper we classify all strongly regular Cayley graphs over this group using Schur rings method. As a consequence we obtain a complete classification of …
Let $G$ denote a finite abelian group with identity 1 and let $S$ denote an inverse-closed subset of $G \setminus {1}$, which generates $G$ and for which there exists $s \in S$, such that $\la S \setminus \{s,s^{-1}\} \ra \ne G$. In this paper we obtain the complete classification of distance-regular Cayley graphs $\cay(G;S)$ for such pairs of $G$ and $S$.
Let $G$ be a generalized dicyclic group with identity $1$. An inverse closed subset $S$ of $G\setminus\{1\}$ is called minimal if $\langle S\rangle=G$ and there exists some $s\in S$ such that $\langle S\setminus\{s,s^{-1}\} \rangle\neq G$. In this paper, we characterize distance-regular Cayley graphs $\mathrm{Cay}(G,S)$ of $G$ under the condition that $S$ is minimal.
In this paper, we give a construction of strongly regular Cayley graphs and a construction of skew Hadamard difference sets. Both constructions are based on choosing cyclotomic classes in finite fields, and they generalize the constructions given by Feng and Xiang \cite{FX111,FX113}. Three infinite families of strongly regular graphs with new parameters are obtained. The main tools that we employed are index 2 Gauss sums, instead of cyclotomic numbers.
In this paper we are interested in the asymptotic enumeration of bipartite Cayley digraphs and Cayley graphs over abelian groups. Let $A$ be an abelian group and let $ι$ be the automorphism of $A$ defined by $a^ι=a^{-1}$, for every $a\in A$. A Cayley graph $\Cay(A, S)$ is said to have an automorphism group as small as possible if $\Aut(\Cay(A,S)) = \langle A,ι\rangle$. In this paper, we show that, except for two infinite families, almost all bipartite Cayley graphs on abelian groups have automorphism group as small as possible. We also investigate the analogous question for bipartite Cayley di…
In this paper, we consider regular automorphism groups of graphs in the RT$2$ family and the Davis-Xiang family and amorphic abelian Cayley schemes from these graphs. We derive general results on the existence of non-abelian regular automorphism groups from abelian regular automorphism groups and apply them to the RT$2$ family and Davis-Xiang family and their amorphic abelian Cayley schemes to produce amorphic non-abelian Cayley schemes.
In \cite{M18}, the first author gave a construction of strongly regular Cayley graphs on the additive group of finite fields by using three-valued Gauss periods. In particular, together with the result in \cite{BLMX}, it was shown that there exists a strongly regular Cayley graph with negative Latin square type parameters $(q^6,r(q^3+1),-q^3+r^2+3r,r^2+r)$, where $r=M(q^2-1)/2$, in the following cases: (i) $M=1$ and $q\equiv 3\,(\mod{4})$; (ii) $M=3$ and $q\equiv 7\,(\mod{24})$; and (iii) $M=7$ and $q\equiv 11,51\,(\mod{56})$. The existence of strongly regular Cayley graphs with the above para…
A well-known conjecture of Alspach says that every $2k$-regular Cayley graph of an abelian group can be decomposed into Hamiltonian cycles. We consider an analogous question for infinite abelian groups. In this setting one natural analogue of a Hamiltonian cycle is a spanning double-ray. However, a naive generalisation of Alspach's conjecture fails to hold in this setting due to the existence of $2k$-regular Cayley graphs with finite cuts $F$ where $|F|$ and $k$ differ in parity, which necessarily preclude the existence of a decomposition into spanning double-rays. We show that every $4$-regul…
Let $\mathcal{M}=CM(D_n,X,p)$ be a regular Cayley map on the dihedral group $D_n$ of order $2n, n \ge 2,$ and let $π$ be the power function associated with $\mathcal{M}$. In this paper it is shown that the kernel Ker$(π)$ of the power function $π$ is a dihedral subgroup of $D_n$ and if $n \ne 3,$ then the kernel Ker$(π)$ is of order at least $4$. Moreover, all $\mathcal{M}$ are classified for which Ker$(π)$ is of order $4$. In particular, besides $4$ sporadic maps on $4,4,8$ and $12$ vertices respectively, two infinite families of non-$t$-balanced Cayley maps on $D_n$ are obtained.
A generalized Petersen graph $GP(n,k)$ is a regular cubic graph on $2n$ vertices (the parameter $k$ is used to define some of the edges). It was previously shown (Ball et al., 2015) that the cop number of $GP(n,k)$ is at most $4$, for all permissible values of $n$ and $k$. In this paper we prove that the cop number of "most" generalized Petersen graphs is exactly $4$. More precisely, we show that unless $n$ and $k$ fall into certain specified categories, then the cop number of $GP(n,k)$ is $4$. The graphs to which our result applies all have girth $8$. In fact, our argument is slightly more ge…
Given a finite simple graph $Γ$ on $n$ vertices its complementary prism is the graph $Γ\barΓ$ that is obtained from $Γ$ and its complement $\barΓ$ by adding a perfect matching, where each its edge connects two copies of the same vertex in $Γ$ and $\barΓ$. It generalizes the Petersen graph, which is obtained if $Γ$ is the pentagon. The automorphism group of $Γ\barΓ$ is described for arbitrary graph $Γ$. In particular, it is shown that the ratio between the cardinalities of the automorphism groups of $Γ\barΓ$ and $Γ$ can attain only values $1$, $2$, $4$, and $12$. It is shown that the Cheeger nu…
The real monomial representations of Clifford algebras give rise to two sequences of bent functions. For each of these sequences, the corresponding Cayley graphs are strongly regular graphs, and the corresponding sequences of strongly regular graph parameters coincide. Even so, the corresponding graphs in the two sequences are not isomorphic, except in the first 3 cases. The proof of this non-isomorphism is a simple consequence of a theorem of Radon.
A subset $R$ of the vertex set of a graph $Γ$ is said to be $(κ,τ)$-regular if $R$ induces a $κ$-regular subgraph and every vertex outside $R$ is adjacent to exactly $τ$ vertices in $R$. In particular, if $R$ is a $(κ,τ)$-regular set of some Cayley graph on a finite group $G$, then $R$ is called a $(κ,τ)$-regular set of $G$. Let $H$ be a non-trivial normal subgroup of $G$, and $κ$ and $τ$ a pair of integers satisfying $0\leqκ\leq|H|-1$, $1\leqτ\leq|H|$ and $\gcd(2,|H|-1)\midκ$. It is proved that (i) if $τ$ is even, then $H$ is a $(κ,τ)$-regular set of $G$; (ii) if $τ$ is odd, then $H$ is a $(κ…
In this paper, we present a method to obtain regular (or equitable) partitions of Cayley (di)graphs (that is, graphs, digraphs, or mixed graphs) of permutation groups on $n$ letters. We prove that every partition of the number $n$ gives rise to a regular partition of the Cayley graph. By using representation theory, we also obtain the complete spectra and the eigenspaces of the corresponding quotient (di)graphs. More precisely, we provide a method to find all the eigenvalues and eigenvectors of such (di)graphs, based on their irreducible representations. As examples, we apply this method to th…
We provide a new derivation of the well-known generating function counting the number of walks on a regular tree that start and end at the same vertex, and more generally, a generating function for the number of walks that end at a vertex a distance i from the start vertex. These formulas seem to be very old, and go back, in an equivalent form, at least to Harry Kesten's work on symmetric random walks on groups from 1959, and in the present form to Brendan McKay (1983).
Let $Γ=\mathrm{Cay}(G,S)$ be a Cayley digraph on a group $G$ and let $A=\mathrm{Aut}(Γ)$. The Cayley index of $Γ$ is $|A:G|$. It has previously been shown that, if $p$ is a prime, $G$ is a cyclic $p$-group and $A$ contains a noncyclic regular subgroup, then the Cayley index of $Γ$ is superexponential in $p$. We present evidence suggesting that cyclic groups are exceptional in this respect. Specifically, we establish the contrasting result that, if $p$ is an odd prime and $G$ is abelian but not cyclic, and has order a power of $p$ at least $p^3$, then there is a Cayley digraph $Γ$ on $G$ whose …
Let $Γ$ be a graph with vertex set $V$, and let $a$ and $b$ be nonnegative integers. A subset $C$ of $V$ is called an $(a,b)$-regular set in $Γ$ if every vertex in $C$ has exactly $a$ neighbors in $C$ and every vertex in $V\setminus C$ has exactly $b$ neighbors in $C$. In particular, $(0, 1)$-regular sets and $(1, 1)$-regular sets in $\Ga$ are called perfect codes and total perfect codes in $\Ga$, respectively. A subset $C$ of a group $G$ is said to be an $(a,b)$-regular set of $G$ if there exists a Cayley graph of $G$ which admits $C$ as an $(a,b)$-regular set. In this paper we prove that, fo…
Commutative Distance-Regularity: Algebraic Hierarchies and Cayley Graph Constructions
Centro de Investigaciones en Ciencias Exactas y Naturales (CIEM), Universidad Nacional de Córdoba（科尔多瓦国立大学精确与自然科学研究中心）；Instituto de Matemática Aplicada San Luis, Universidad Nacional de San Luis and CONICET（圣路易斯应用数学研究所，圣路易斯国立大学与阿根廷国家科学研究委员会）；Departamento de Matemática, Universidad Nacional de San Luis（圣路易斯国立大学数学系）
In a graph $Γ$ with vertex set $V$, a subset $C$ of $V$ is called an $(a,b)$-perfect set if every vertex in $C$ has exactly $a$ neighbors in $C$ and every vertex in $V\setminus C$ has exactly $b$ neighbors in $C$, where $a$ and $b$ are nonnegative integers. In the literature $(0,1)$-perfect sets are known as perfect codes and $(1,1)$-perfect sets are known as total perfect codes. In this paper we prove that, for any finite group $G$, if a non-trivial normal subgroup $H$ of $G$ is a perfect code in some Cayley graph of $G$, then it is also an $(a,b)$-perfect set in some Cayley graph of $G$ for …
The concept of directed strongly regular graphs (DSRG) was introduced by Duval in 1988 \cite{A}.In the present paper,we use representation theory of finite groups in order to investigate the directed strongly regular Cayley graphs.We first show that a Cayley graph $\mathcal{C}(G,S)$ is not a directed strongly regular graph if $S$ is a union of some conjugate classes of $G$.This generalizes an earlier result of Leif K.Jørgensen \cite{J1} on abelian groups.Secondly,by using induced representations,we have a look at the Cayley graph $\mathcal{C}(N\rtimes_θH, N_1\times H_1)$ with $N_1\subseteq N$ …
It was proved in [Y.-Q. Feng, C. H. Li and J.-X. Zhou, Symmetric cubic graphs with solvable automorphism groups, {\em European J. Combin.} {\bf 45} (2015), 1-11] that a cubic symmetric graph with a solvable automorphism group is either a Cayley graph or a $2$-regular graph of type $2^2$, that is, a graph with no automorphism of order $2$ interchanging two adjacent vertices. In this paper an infinite family of non-Cayley cubic $2$-regular graphs of type $2^2$ with a solvable automorphism group is constructed. The smallest graph in this family has order 6174.
