Reaction Graphs

Chemical reaction graphs (for a fixed type of rearrangement) are orbital graphs for transitive permutation representations of symmetric groups, so algebraic combinatorics and group theory are effective tools for studying such properties as their connectivity and automorphisms. For example, we construct orbital graphs (and, hence, reaction graphs) from Cayley diagrams by contracting edges, and use graph-embeddings in surfaces to determine the automorphism groups of these graphs. We apply these ideas to the rearrangements of the P ₇ ^3- -ion and of bullvalene, together with some purely mathematical examples of reaction graphs.     

Partition graphs for finite symmetric groups

This paper outlines an investigation of a class of arc-transitive graphs admitting a finite symmetric group S_n acting primitively on vertices, with vertex-stabilizer isomorphic to the wreath product S_m wr S_r (preserving a partition of {1,2,…n} into r parts of equal size m). Several properties of these graphs are considered, including their correspondence with r × r matrices with constant row- and column-sums equal to m, their girth, and the local action of the vertex-stabilizer. Also, it is shown that the only instance where S_n acts transitively on 2-arcs occurs in the case m = r = 2. © 1997 John Wiley & Sons, Inc. J Graph Theory 25: 107–117, 1997     

On the extendability of certain semi-Cayley graphs of finite abelian groups

A connected graph Γ with at least 2n+2 vertices is said to be n-extendable if every matching of size n in Γ can be extended to a perfect matching. The aim of this paper is to study the 1-extendability and 2-extendability of certain semi-Cayley graphs of finite abelian groups, and the classification of connected 2-extendable semi-Cayley graphs of finite abelian groups is given. Thus the 1-extendability and 2-extendability of Cayley graphs of non-abelian groups which can be realized as such semi-Cayley graphs of abelian groups can be deduced. In particular, the 1-extendability and 2-extendability of connected Cayley graphs of generalized dicyclic groups and generalized dihedral groups are characterized.     

Half-Transitive Graphs of Prime-Cube Order

We call an undirected graph X half-transitive if the automorphism group Aut X of X acts transitively on the vertex set and edge set but not on the set of ordered pairs of adjacent vertices of X. In this paper we determine all half-transitive graphs of order p ³ and degree 4, where p is an odd prime; namely, we prove that all such graphs are Cayley graphs on the non-Abelian group of order p ³ and exponent p ², and up to isomorphism there are exactly (p – 1)/2 such graphs. As a byproduct, this proves the uniqueness of Holt's half-transitive graph with 27 vertices.     

On transitive one-factorizations of arc-transitive graphs

It is shown that transitive 1-factorizations of arc-transitive graphs exist if and only if certain factorizations of their automorphism groups exist. This relation provides a method for constructing and characterizing transitive 1-factorizations for certain classes of arc-transitive graphs. Then a characterization is given of 2-arc-transitive graphs which have transitive 1-factorizations. In this characterization, some 2-arc transitive graphs and their transitive 1-factorizations are constructed.     

Planarity of Cayley graphs of graph products of groups

We obtain a complete classification of graph products of finite abelian groups whose Cayley graphs with respect to the standard presentations are planar.     

关于交换群上的Cayley有向图的正规性

Cayley有向图X=Cay(G,S)叫做正规的,如果G的右正则表示R(G)在X的全自同构群Aut(X)中正规,我们定出了交换群上的小度数的非正规的Cayley有向图, 并给出了一个猜想．应用这个结果,给出了pn(n≤2)个点上的度数不超过3的有向对称图的分类,这里p是一个奇素数．     

Characterizations and recognition of circular-arc graphs and subclasses: A survey

Circular graphs are intersection graphs of arcs on a circle. These graphs are reported to have been studied since 1964, and they have been receiving considerable attention since a series of papers by Tucker in the 1970s. Various subclasses of circular-arc graphs have also been studied. Among these are the proper circular-arc graphs, unit circular-arc graphs, Helly circular-arc graphs and co-bipartite circular-arc graphs. Several characterizations and recognition algorithms have been formulated for circular-arc graphs and its subclasses. In particular, it should be mentioned that linear time algorithms are known for all these classes of graphs. In the present paper, we survey these characterizations and recognition algorithms, with emphasis on the linear time algorithms.     

The locally 2-arc transitive graphs admitting an almost simple group of Suzuki type

In this paper, seven families of vertex-intransitive locally (G,2)-arc transitive graphs are constructed, where Sz(q)?G?Aut(Sz(q)), q=²2k+1 for some k∈N. It is then shown that for any graph Γ in one of these families, Sz(q)?Aut(Γ)?Aut(Sz(q)) and that the only locally 2-arc transitive graphs admitting an almost simple group of Suzuki type whose vertices all have valency at least three are (i) graphs in these seven families, (ii) (vertex transitive) 2-arc transitive graphs admitting an almost simple group of Suzuki type, or (iii) double covers of the graphs in (ii). Since the graphs in (ii) have been classified by Fang and Praeger (1999) [6], this completes the classification of locally 2-arc transitive graphs admitting a Suzuki simple group     

Completions of Goldschmidt Amalgams of Type G4 in Dimension 3

The subgroups of GL₃(k) which are completions of the Goldschmidt G ₄-amalgam are determined. We also draw attention to five related graphs which are remarkable in that they have large girth and few vertices.     

Invariant Kekulé structures in fullerene graphs

Fullerene graphs are trivalent plane graphs with only hexagonal and pentagonal faces. They are often used to model large carbon molecules: each vertex represents a carbon atom and the edges represent chemical bonds. A totally symmetric Kekulé structure in a fullerene graph is a set of independent edges which is fixed by all symmetries of the fullerene and molecules with totally symmetric Kekulé structures could have special physical and chemical properties, as suggested in [Austin, S.J, and J. Baker, P. W. Fowler, D. E. Manolopoulos, Bond-stretch Isomerism and the Fullerenes, J. Chem. Soc. Perkin Trans. 2 (1994), 2319–2323] and [Rogers, K.M., and P. W. Fowler, Leapfrog fullerenes, Huckel bond order and Kekulé structures, J. Chem. Soc. Perkin Trans. 2 (2001), 18–22]. All fullerenes with at least ten symmetries were studied in [Graver, J.E. The Structure of Fullerene Signature, DIMACS Series of Discrete Mathematics and Theoretical Computer Science 64, AMS (2005), 137–166.] and a complete catalog was given in [Graver, J. E. Catalog of All Fullerene with Ten or More Symmetries DIMACS Series of Discrete Mathematics and Theoretical Computer Science 64 AMS (2005), 167–188]. Starting from this catalog in [Bogaerts, M., and G. Mazzuoccolo, G.Rinaldi, Totally symmetric Kekulé structures in fullerene graphs with ten or more symmetries, MATCH Communications in Mathematical and in Computer Chemistry 69 (2013), 677–705] we established exactly which of them have at least one totally symmetric Kekulé structure.     

Two-ended regular median graphs

We show that regular median graphs of linear growth are the Cartesian product of finite hypercubes with the two-way infinite path. Such graphs are Cayley graphs and have only two ends.For cubic median graphs G the condition of linear growth can be weakened to the condition that G has two ends. For higher degree the relaxation to two-ended graphs is not possible, which we demonstrate by an example of a median graph of degree four that has two ends, but nonlinear growth.     

Forbidden induced subgraph characterizations of subclasses and variations of perfect graphs: A survey

A graph is perfect if the chromatic number is equal to the clique number for every induced subgraph of the graph. Perfect graphs were defined by Berge in the sixties. In this survey we present known results about partial characterizations by forbidden induced subgraphs of different graph classes related to perfect graphs. We analyze a variation of perfect graphs, clique-perfect graphs, and two subclasses of perfect graphs, coordinated graphs and balanced graphs.     

Marginal AMP chain graphs

We present a new family of models that is based on graphs that may have undirected, directed and bidirected edges. We name these new models marginal AMP (MAMP) chain graphs because each of them is Markov equivalent to some AMP chain graph under marginalization of some of its nodes. However, MAMP chain graphs do not only subsume AMP chain graphs but also multivariate regression chain graphs. We describe global and pairwise Markov properties for MAMP chain graphs and prove their equivalence for compositional graphoids. We also characterize when two MAMP chain graphs are Markov equivalent.For Gaussian probability distributions, we also show that every MAMP chain graph is Markov equivalent to some directed and acyclic graph with deterministic nodes under marginalization and conditioning on some of its nodes. This is important because it implies that the independence model represented by a MAMP chain graph can be accounted for by some data generating process that is partially observed and has selection bias. Finally, we modify MAMP chain graphs so that they are closed under marginalization for Gaussian probability distributions. This is a desirable feature because it guarantees parsimonious models under marginalization.     

On conservative and supermagic graphs

A graph is called supermagic if it admits a labelling of the edges by pairwise different consecutive positive integers such that the sum of the labels of the edges incident with a vertex is independent of the particular vertex. A graph G is called conservative if it admits an orientation and a labelling of the edges by integers {1,…,|E(G)|} such that at each vertex the sum of the labels on the incoming edges is equal to the sum of the labels on the outgoing edges. In this paper we deal with conservative graphs and their connection with the supermagic graphs. We introduce a new method to construct supermagic graphs using conservative graphs. Inter alia we show that the union of some circulant graphs and regular complete multipartite graphs are supermagic.     

Polar cographs

Polar graphs are a natural extension of some classes of graphs like bipartite graphs, split graphs and complements of bipartite graphs. A graph is (s,k)-polar if there exists a partition A,B of its vertex set such that A induces a complete s-partite graph (i.e., a collection of at most s disjoint stable sets with complete links between all sets) and B a disjoint union of at most k cliques (i.e., the complement of a complete k-partite graph).Recognizing a polar graph is known to be NP-complete. These graphs have not been extensively studied and no good characterization is known. Here we consider the class of polar graphs which are also cographs (graphs without induced path on four vertices). We provide a characterization in terms of forbidden subgraphs. Besides, we give an algorithm in time O(n) for finding a largest induced polar subgraph in cographs; this also serves as a polar cograph recognition algorithm. We examine also the monopolar cographs which are the (s,k)-polar cographs where min(s,k)?1. A characterization of these graphs by forbidden subgraphs is given. Some open questions related to polarity are discussed.     

On 2‐factorizations of the complete graph: From the k‐pyramidal to the universal property

We consider 2‐factorizations of complete graphs that possess an automorphism group fixing k?0 vertices and acting sharply transitively on the others. We study the structures of such factorizations and consider the cases in which the group is either abelian or dihedral in some more details. Combining results of the first part of the paper with a result of D. Bryant, J Combin Des, 12 (2004), 147–155, we prove that the class of 2‐factorizations of complete graphs is universal. Namely each finite group is the full automorphism group of a 2‐factorization of the class. © 2009 Wiley Periodicals, Inc. J Combin Designs 17: 211‐228, 2009     

On minimal forbidden subgraph characterizations of balanced graphs

A graph is balanced if its clique-matrix contains no edge–vertex incidence matrix of an odd chordless cycle as a submatrix. While a forbidden induced subgraph characterization of balanced graphs is known, there is no such characterization by minimal forbidden induced subgraphs. In this work, we provide minimal forbidden induced subgraph characterizations of balanced graphs restricted to graphs that belong to one of the following graph classes: complements of bipartite graphs, line graphs of multigraphs, and complements of line graphs of multigraphs. These characterizations lead to linear-time recognition algorithms for balanced graphs within the same three graph classes.