Permutation Groups with a Cyclic Regular Subgroup and Arc Transitive Circulants

We classify the nite primitive permutation groups which have a cyclic subgroup with two orbits. This extends classical topics in permutation group theory, and has arithmetic consequences. By a theorem of C. L. Siegel, ane algebraic curves with innitely many integral points are parametrized by rational functions whose monodromy groups have this property. We classify the possibilities of these monodromy groups, and give applications to Hilbert’s irreducibility theorem.

Problem statement: The number of spanning trees τ(G) in graphs (networks) is an important invariant, it is also an important measure of reliability of a network. Approach: Using linear algebra and matrix analysis techniques to evaluate the associated determinants. Results: In this study we derive simple formulas for the number of spanning trees of complete graph Kn and complete bipartite graph Kn,m and some of their applications. A large number of theorems of number of the spanning trees of known operations on complete graph Kn and complete bipartite graph Kn,m are obtained. Conclusion: The evaluation of number of spanning trees is not only interesting from a mathematical (computational) perspective, but also, it is an important measure of reliability of a network and designing electrical circuits. Some computationally hard problems such as the travelling salesman problem can be solved approximately by using spanning trees. Due to the high dependence of the network design and reliability on the graph theory we introduced the following important theorems and lemmas and their proofs.

A proof of a conjecture of Sabidussi on graphs idempotent under the lexicographic product

A finite permutation group [Formula: see text] is called [Formula: see text]-closed if [Formula: see text] is the largest subgroup of [Formula: see text] which leaves invariant each of the [Formula: see text]-orbits for the induced action on [Formula: see text]. Introduced by Wielandt in 1969, the concept of [Formula: see text]-closure has developed as one of the most useful approaches for studying relations on a finite set invariant under a group of permutations of the set; in particular for studying automorphism groups of graphs and digraphs. The concept of total [Formula: see text]-closure switches attention from a particular group action, and is a property intrinsic to the group: a finite group [Formula: see text] is said to be totally [Formula: see text]-closed if [Formula: see text] is [Formula: see text]-closed in each of its faithful permutation representations. There are infinitely many finite soluble totally [Formula: see text]-closed groups, and these have been completely characterized, but up to now no insoluble examples were known. It turns out, somewhat surprisingly to us, that there are exactly [Formula: see text] totally [Formula: see text]-closed finite nonabelian simple groups: the Janko groups [Formula: see text], [Formula: see text] and [Formula: see text], together with [Formula: see text], [Formula: see text] and the Monster [Formula: see text]. Moreover, if a finite totally [Formula: see text]-closed group has no nontrivial abelian normal subgroup, then we show that it is a direct product of some (but not all) of these simple groups, and there are precisely [Formula: see text] examples. In the course of obtaining this classification, we develop a general framework for studying [Formula: see text]-closures of transitive permutation groups, which we hope will prove useful for investigating representations of finite groups as automorphism groups of graphs and digraphs, and in particular for attacking the long-standing polycirculant conjecture. In this direction, we apply our results, proving a dual to a 1939 theorem of Frucht from Algebraic Graph Theory. We also pose several open questions concerning closures of permutation groups.

A group G is called a \({\mathcal {T}_{c}}\)-group if every cyclic subnormal subgroup of G is normal in G. Similarly, classes \({\mathcal {PT}_{c}}\) and \({\mathcal {PST}_{c}}\) are defined, by requiring cyclic subnormal subgroups to be permutable or S-permutable, respectively. A subgroup H of a group G is called normal (permutable or S-permutable) cyclic sensitive if whenever X is a normal (permutable or S-permutable) cyclic subgroup of H there is a normal (permutable or S-permutable) cyclic subgroup Y of G such that \({X=Y \cap H}\). We analyze the behavior of a collection of cyclic normal, permutable and S-permutable subgroups under the intersection map into a fixed subgroup of a group. In particular, we tie the concept of normal, permutable and S-permutable cyclic sensitivity with that of \({\mathcal {T}_c}\), \({\mathcal {PT}_c}\) and \({\mathcal {PST}_c}\) groups. In the process we provide another way of looking at Dedekind, Iwasawa and nilpotent groups.

Perfect domination sets in Cayley graphs

Graph decompositions are vital in the study of combinatorial design theory. A decomposition of a graph G is a partition of its edge set. An n-sun graph is a cycle Cn with an edge terminating in a vertex of degree one attached to each vertex. In this paper, we define n-sun decomposition of some even order graphs with a perfect matching. We have proved that the complete graph K2n, complete bipartite graph K2n, 2n and the Harary graph H4, 2n have n-sun decompositions. A labeling scheme is used to construct the n- suns.

Let n≥2 be an integer. The complete graph Kn with a 1-factor F removed has a decomposition into Hamilton cycles if and only if n is even. We show that Kn−F has a decomposition into Hamilton cycles which are symmetric with respect to the 1-factor F if and only if n≡2, 4 mod 8. We also show that the complete bipartite graph Kn, n has a symmetric Hamilton cycle decomposition if and only if n is even, and that if F is a 1-factor of Kn, n, then Kn, n−F has a symmetric Hamilton cycle decomposition if and only if n is odd. © 2010 Wiley Periodicals, Inc. J Combin Designs 19:1-15, 2010

Computing in permutation and matrix groups III: Sylow subgroups

Supposed G be a simple graph with a set of vertices (G), set of edges E(G), with | E(G)| = q. A graph G is a harmonious graph if there is an injective function f*: (G) → ℤq, such that the induced function f*: E (G) → ℤq defined by f*(xy) = f(x) + f(y), ∀xy ∈ E(G) is a bijective function. The function f is called harmonious labeling of G. It was known that a complete graph Kn is harmonious only for n ≤ 4 In this paper, we investigate the existence of harmonious labeling of the graphs from the corona operation between a complete graph K4 and Kn Kn¯, and also between K5 and Kn¯

By JOHANNES SIEMONS and ASCHER WAGNER 1. Introduction. A permutation group G acting on a set f2 induces a permutation group on the unordered sets of k distinct points. If H is another permutation group on f2 we shall write H ~ G if H and G have the same orbits on the unordered sets of k points. Bercov and Hobby [2] have shown that for infinite groups H k G implies H ,,~ G if I < k. In [9] we have shown that this result is also true for finite groups, with the obviously necessary condition that k < 89 f2l- In [9] it is also shown that for finite groups H 2 G implies that H and G are either both primitive or both imprimitive with the same blocks of imprimi- tivity. If H and G have the same orbits on all subsets of ~2 we shall say that H and G are orbit equivalent and write H ~ G. Orbit equivalence for groups acting on quite arbitrary f2 has been considered by Betten [3]; the main results concern intransitive groups. In this paper we shall be concerned only with orbit equivalence for finite groups. In this case, of course, H ~ G if, and only if, H k G for all k. Suppose that H ~ G and that L is a permutation group on a set A. Then the direct products H x L and G x L, acting naturally on f2 u A, are orbit equivalent and intransi- tive. Also, if L is transitive on A the wreath products H 2, L and G 2, L, acting naturally on the direct sum of [A h copies of f2, are orbit equivalent and imprimitive. This suggests that the basic situation to investigate is when G, hence also H, is primitive on f2. Without loss of generality we may assume that H c G since H ~ G implies H ~ (H, G). Our main result is the following theorem. Theorem A. Let K be a finite primitive permutation group on a set f2. Let H c K and H ,~ K. Suppose there exists aprime r dividing the order of K but not the order of H. Then only the following possibilities exist: H K I g21 r 2-sets 3-sets 4-sets (i) ~3 5;3 3 2 (i 0 C 5 Dlo 5 2 5; 5 (iii) A~ (5) S s 5 3 10 (iv) A x (8) FA~ (S) 8 3 28 (V) A 1 (8) 23 \\PSL 3 (2) 8 3 28 (vi)

On sequences of polynomials arising from graph invariants

Metric dimension and pattern avoidance in graphs

The present article presents some new results relating to Atomic Bond Connectivity energies and Spectral radii of generalized splitting and generalized shadow graphs constructed on the basis of some fundamental families of cycle graph Cn, complete graph Kn and complete bipartite graph Kn,n referred as base graphs. In fact we relate the energies and Spectral radii of splitting and shadow graphs with the energies and Spectral radii of original graphs.

Let t(G) be the number of spanning trees of a connected graph G, and let b(G) be the number of bases of the bicircular matroid B(G). In this paper we obtain bounds relating b(G) and t(G), and study in detail the case where G is a complete graph Kn or a complete bipartite graph Kn,m.