A switching for all strongly regular collinearity graphs from polar spaces

We describe a general construction of strongly regular graphs from the collinearity graph of a finite classical polar spaces of rank at least 3 over a finite field of order q. We show that these graphs are non-isomorphic to the collinearity graphs and have the same parameters. For most of these parameters, the collinearity graphs were the only known examples, and so many of our examples are new.     

Computation of best possible low degree expanders

We present an algorithm for computing a best possible bipartite cubic expander for a given number of vertices. Such graphs are needed in many applications and are also the basis for many results in theoretical computer science. Known construction methods for expander graphs yield expanders that have a fairly poor expansion compared to the best possible expansion. Our algorithm is based on a lemma which allows to calculate an upper bound for the expansion of cubic bipartite graphs.     

Geometric aspects of 2-walk-regular graphs

A t-walk-regular graph is a graph for which the number of walks of given length between two vertices depends only on the distance between these two vertices, as long as this distance is at most t. Such graphs generalize distance-regular graphs and t-arc-transitive graphs. In this paper, we will focus on 1- and in particular 2-walk-regular graphs, and study analogues of certain results that are important for distance-regular graphs. We will generalize Delsarte?s clique bound to 1-walk-regular graphs, Godsil?s multiplicity bound and Terwilliger?s analysis of the local structure to 2-walk-regular graphs. We will show that 2-walk-regular graphs have a much richer combinatorial structure than 1-walk-regular graphs, for example by proving that there are finitely many non-geometric 2-walk-regular graphs with given smallest eigenvalue and given diameter (a geometric graph is the point graph of a special partial linear space); a result that is analogous to a result on distance-regular graphs. Such a result does not hold for 1-walk-regular graphs, as our construction methods will show.     

New Classes of Counterexamples to Hendrickson’s Global Rigidity Conjecture

We examine the generic local and global rigidity of various graphs in ℝ ^d . Bruce Hendrickson showed that some necessary conditions for generic global rigidity are (d+1)-connectedness and generic redundant rigidity, and hypothesized that they were sufficient in all dimensions. We analyze two classes of graphs that satisfy Hendrickson’s conditions for generic global rigidity, yet fail to be generically globally rigid. We find a large family of bipartite graphs for d>3, and we define a construction that generates infinitely many graphs in ℝ⁵. Finally, we state some conjectures for further exploration.     

Trees with very few eigenvalues

The number of distinct eigenvalues of the adjacency matrix of a graph is bounded below by the diameter of the graph plus one. Many graphs that achieve this lower bound exhibit much symmetry, for example, distance-transitive and distance-regular graphs. Here we provide a recursive construction that will create graphs having the fewest possible eigenvalues. This construction is best at creating trees, but will also create cyclic graphs meeting the lower bound. Unlike the graphs mentioned above, many of the graphs constructed do not exhibit large amounts of symmetry. A corollary allows us to determine the values and multiplicities of all the nonsimple eigenvalues of the constructed graph.     

Spectra of coronae

We introduce a new invariant, the coronal of a graph, and use it to compute the spectrum of the corona G°H of two graphs G and H. In particular, we show that this spectrum is completely determined by the spectra of G and H and the coronal of H. Previous work has computed the spectrum of a corona only in the case that H is regular. We then explicitly compute the coronals for several families of graphs, including regular graphs, complete n-partite graphs, and paths. Finally, we use the corona construction to generate many infinite families of pairs of cospectral graphs.     

End-transitive graphs

We investigate the structure of connected graphs, not necessarily locally finite, with infinitely many ends. We study end-transitive such graphs and such graphs with the property that the stabilizer of some end acts transitively on the vertices of the graph. In both cases we show that the graphs have a tree-like structure.     

Cayley Cages

A (k,g)-Cayley cage is a k-regular Cayley graph of girth g and smallest possible order. We present an explicit construction of (k,g)-Cayley graphs for all parameters k≥2 and g≥3 and generalize this construction to show that many well-known small k-regular graphs of girth g can be constructed in this way. We also establish connections between this construction and topological graph theory, and address the question of the order of (k,g)-Cayley cages.     

A New Method for Enumerating Independent Sets of a Fixed Size in General Graphs

We develop a new method for enumerating independent sets of a fixed size in general graphs, and we use this method to show that a conjecture of Engbers and Galvin [7] holds for all but finitely many graphs. We also use our method to prove special cases of a conjecture of Kahn [13]. In addition, we show that our method is particularly useful for computing the number of independent sets of small sizes in general regular graphs and Moore graphs, and we argue that it can be used in many other cases when dealing with graphs that have numerous structural restrictions.     

Lifting constructions of strongly regular Cayley graphs

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 sets. The second construction uses quadratic forms over finite fields and it is a common generalization of the construction of the affine polar graphs [7] and a construction of strongly regular Cayley graphs given in [15]. 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 obtained from the first construction can serve as starters for the second construction. We also obtain association schemes from the second construction.     

Topological realizations of ortho-projection graphs

We discuss the production of ortho-projection graphs from alternating knot diagrams, and introduce a more general construction of such graphs from “splittings” of closed, non-orientable surfaces. As our main result, we prove that this new topological construction generates all ortho-projection graphs. We present a minimal example of an ortho-projection graph that does not arise from a knot diagram, and provide a surface-splitting that realizes this graph.     

Vertex-magic labelings of regular graphs II

Previously the first author has shown how to construct vertex-magic total labelings (VMTLs) for large families of regular graphs. The construction proceeds by successively adding arbitrary 2-factors to a regular graph of order n which possesses a strong VMTL, to produce a regular graph of the same order but larger size. In this paper, we exploit this construction method. We are able to show that for any r≥4, every r-regular graph of odd order n≤17 has a strong VMTL. We show how to produce strong labelings for some families of 2-regular graphs since these are used as the starting points of our construction. While even-order regular graphs are much harder to deal with, we introduce ‘mirror’ labelings which provide a suitable starting point from which the construction can proceed. We are able to show that several large classes of r-regular graphs of even order (including some Hamiltonian graphs) have VMTLs.     

Eigenvalues of graphs with vertices of large degree at distance three apart

We investigate the family of graphs with many large eigenvalues. It is not hard to see that every graph with many vertices of large degree that are pairwise at distance at least four from each other, has many large eigenvalues. We show that this does not hold if the vertices of large degree are at mutual distance three from each other. We explore this class of graphs further and provide some bounds on their eigenvalues.     

Pairs of Chromatically Equivalent Graphs

Two graphs are said to be chromatically equivalent if they have the same chromatic polynomial. In this paper we give the means to construct infinitely many pairs of chromatically equivalent graphs where one graph in the pair is clique-separable, that is, can be obtained by identifying an r-clique in some graph H ₁ with an r-clique in some graph H ₂, and the other graph is non-clique-separable. There are known methods for finding pairs of chromatically equivalent graphs where both graphs are clique-separable or both graphs are non-clique-separable. Although examples of pairs of chromatically equivalent graphs where only one of the graphs is clique-separable are known, a method for the construction of infinitely many such pairs was not known. Our method constructs such pairs of graphs with odd order n ≥ 9.     

Enumeration problems for classes of self-similar graphs

We describe a general construction principle for a class of self-similar graphs. For various enumeration problems, we show that this construction leads to polynomial systems of recurrences and provide methods to solve these recurrences asymptotically. This is shown for different examples involving classical self-similar graphs such as the Sierpiński graphs. The enumeration problems we investigate include counting independent subsets, matchings and connected subsets.     

Automorphisms of graphs and coverings

We apply the theory of covering spaces to show how one can construct infinitely many finite s-transitive or locally s-transitive graphs. N. Biggs has used for similar purpose a special graph covering construction due to J. H. Conway.     

Connectedness of Efficient Solutions in Multiple Objective Combinatorial Optimization

Connectedness of efficient solutions is a powerful property in multiple objective combinatorial optimization since it allows the construction of the complete efficient set using neighborhood search techniques. However, we show that many classical multiple objective combinatorial optimization problems do not possess the connectedness property in general, including, among others, knapsack problems (and even several special cases) and linear assignment problems. We also extend known non-connectedness results for several optimization problems on graphs like shortest path, spanning tree and minimum cost flow problems. Different concepts of connectedness are discussed in a formal setting, and numerical tests are performed for two variants of the knapsack problem to analyze the likelihood with which non-connected adjacency graphs occur in randomly generated instances.     

Constructing isospectral non‐isomorphic digraphs from hypergraphs

We explore the “oriented line graph” construction associated with a hypergraph, leading to a construction of pairs of strongly connected directed graphs whose adjacency operators have the same spectra. We give conditions on a hypergraph so that a hypergraph and its dual give rise to isospectral, but non‐isomorphic, directed graphs. The proof of isospectrality comes from an argument centered around hypergraph zeta functions as defined by Storm. To prove non‐isomorphism, we establish a Whitney‐type result by showing that the oriented line graphs are isomorphic if and only if the hypergraphs are. © 2009 Wiley Periodicals, Inc. J Graph Theory 63: 231–242, 2010     

On the energy of some circulant graphs

We give an explicit construction of circulant graphs of very high energy. This construction is based on Gauss sums. We also show the Littlewood conjecture can be used to establish new result for a certain class of circulant graphs.     

Recursive constructions of small regular graphs of given degree and girth

We revisit and generalize a recursive construction due to Sachs involving two graphs which increases the girth of one graph and the degree of the other. We investigate the properties of the resulting graphs in the context of cages and construct families of small graphs using geometric graphs, Paley graphs, and techniques from the theory of Cayley maps.