On almost distance-regular graphs

Distance-regular graphs are a key concept in Algebraic Combinatorics and have given rise to several generalizations, such as association schemes. Motivated by spectral and other algebraic characterizations of distance-regular graphs, we study ‘almost distance-regular graphs’. We use this name informally for graphs that share some regularity properties that are related to distance in the graph. For example, a known characterization of a distance-regular graph is the invariance of the number of walks of given length between vertices at a given distance, while a graph is called walk-regular if the number of closed walks of given length rooted at any given vertex is a constant. One of the concepts studied here is a generalization of both distance-regularity and walk-regularity called m-walk-regularity. Another studied concept is that of m-partial distance-regularity or, informally, distance-regularity up to distance m. Using eigenvalues of graphs and the predistance polynomials, we discuss and relate these and other concepts of almost distance-regularity, such as their common generalization of (?,m)-walk-regularity. We introduce the concepts of punctual distance-regularity and punctual walk-regularity as a fundament upon which almost distance-regular graphs are built. We provide examples that are mostly taken from the Foster census, a collection of symmetric cubic graphs. Two problems are posed that are related to the question of when almost distance-regular becomes whole distance-regular. We also give several characterizations of punctually distance-regular graphs that are generalizations of the spectral excess theorem.     

Connected graphs with maximal Q-index: The one-dominating-vertex case

By the signless Laplacian of a (simple) graph G we mean the matrix Q(G)=D(G)+A(G), where A(G),D(G) denote respectively the adjacency matrix and the diagonal matrix of vertex degrees of G. For every pair of positive integers n,k, it is proved that if 3?k?n-3, then H_n,k, the graph obtained from the star K_1,n-1 by joining a vertex of degree 1 to k+1 other vertices of degree 1, is the unique connected graph that maximizes the largest signless Laplacian eigenvalue over all connected graphs with n vertices and n+k edges.     

Feasibility conditions for the existence of walk-regular graphs

A graph X is walk-regular if the vertex-deleted subgraphs of X all have the same characteristic polynomial. Examples of such graphs are vertex-transitive graphs and distance-regular graphs. We show that the usual feasibility conditions for the existence of a distance-regular graph with a given intersection array can be extended so that they apply to walk-regular graphs. Despite the greater generality, our proofs are more elementary than those usually given for distance-regular graphs. An application to the computation of vertex-transitive graphs is described.     

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.     

Amply regular graphs and block designs

We study the amply regular diameter d graphs Γ such that for some vertex a the set of vertices at distance d from a is the set of points of a 2-design whose set of blocks consists of the intersections of the neighborhoods of points with the set of vertices at distance d-1 from a. We prove that the subgraph induced by the set of points is a clique, a coclique, or a strongly regular diameter 2 graph. For diameter 3 graphs we establish that this construction is a 2-design for each vertex a if and only if the graph is distance-regular and for each vertex a the subgraph Γ₃(a) is a clique, a coclique, or a strongly regular graph. We obtain the list of admissible parameters for designs and diameter 3 graphs under the assumption that the subgraph induced by the set of points is a Seidel graph. We show that some of the parameters found cannot correspond to distance-regular graphs.     

A Generalization of Opsut’s Lower Bounds for the Competition Number of a Graph

The notion of a competition graph was introduced by Cohen in 1968. The competition graph C(D) of a digraph D is a (simple undirected) graph which has the same vertex set as D and has an edge between two distinct vertices x and y if and only if there exists a vertex v in D such that (x, v) and (y, v) are arcs of D. For any graph G, G together with sufficiently many isolated vertices is the competition graph of some acyclic digraph. In 1978, Roberts defined the competition number k(G) of a graph G as the minimum number of such isolated vertices. In general, it is hard to compute the competition number k(G) for a graph G and it has been one of the important research problems in the study of competition graphs to characterize a graph by its competition number. In 1982, Opsut gave two lower bounds for the competition number of a graph. In this paper, we give a generalization of these two lower bounds for the competition number of a graph.     

The competition number of a graph and the dimension of its hole space

The competition graph of a digraph D is a (simple undirected) graph which has the same vertex set as D and has an edge between x and y if and only if there exists a vertex v in D such that (x,v) and (y,v) are arcs of D. For any graph G, G together with sufficiently many isolated vertices is the competition graph of some acyclic digraph. The competition number k(G) of G is the smallest number of such isolated vertices. In general, it is hard to compute the competition number k(G) for a graph G and it has been one of the important research problems in the study of competition graphs to characterize a graph by its competition number. Recently, the relationship between the competition number and the number of holes of a graph has been studied. A hole of a graph is a cycle of length at least 4 as an induced subgraph. In this paper, we conjecture that the dimension of the hole space of a graph is not smaller than the competition number of the graph. We verify this conjecture for various kinds of graphs and show that our conjectured inequality is indeed an equality for connected triangle-free graphs.     

The elimination procedure for the competition number is not optimal

Given an acyclic digraph D, the competition graph C(D) is defined to be the undirected graph with V(D) as its vertex set and where vertices x and y are adjacent if there exists another vertex z such that the arcs (x,z) and (y,z) are both present in D. The competition number k(G) for an undirected graph G is the least number r such that there exists an acyclic digraph F on |V(G)|+r vertices where C(F) is G along with r isolated vertices. Kim and Roberts [The Elimination Procedure for the Competition Number, Ars Combin. 50 (1998) 97-113] introduced an elimination procedure for the competition number, and asked whether the procedure calculated the competition number for all graphs. We answer this question in the negative by demonstrating a graph where the elimination procedure does not calculate the competition number. This graph also provides a negative answer to a similar question about the related elimination procedure for the phylogeny number introduced by the current author in [S.G. Hartke, The Elimination Procedure for the Phylogeny Number, Ars Combin. 75 (2005) 297-311].     

Two distance-regular graphs

We construct two families of distance-regular graphs, namely the subgraph of the dual polar graph of type B ₃(q) induced on the vertices far from a fixed point, and the subgraph of the dual polar graph of type D ₄(q) induced on the vertices far from a fixed edge. The latter is the extended bipartite double of the former.     

The competition number of a graph whose holes do not overlap much

Let D be an acyclic digraph. The competition graph of D is a graph which has the same vertex set as D and has an edge between u and v if and only if there exists a vertex x in D such that (u,x) and (v,x) are arcs of D. For any graph G, G together with sufficiently many isolated vertices is the competition graph of some acyclic digraph. The competition number k(G) of G is the smallest number of such isolated vertices.A hole of a graph is an induced cycle of length at least four. Kim (2005) [8] conjectured that the competition number of a graph with h holes is at most h+1. Recently, Li and Chang (2009) [11] showed that the conjecture is true when the holes are independent. In this paper, we show that the conjecture is true though the holes are not independent but mutually edge-disjoint.     

On the connectedness of the complement of a ball in distance-regular graphs

An important property of strongly regular graphs is that the second subconstituent of any primitive strongly regular graph is always connected. Brouwer asked to what extent this statement can be generalized to distance-regular graphs. In this paper, we show that if γ is any vertex of a distance-regular graph Γ and t is the index where the standard sequence corresponding to the second largest eigenvalue of Γ changes sign, then the subgraph induced by the vertices at distance at least t from γ, is connected.     

Whiskers and sequentially Cohen-Macaulay graphs

We investigate how to modify a simple graph G combinatorially to obtain a sequentially Cohen-Macaulay graph. We focus on adding configurations of whiskers to G, where to add a whisker one adds a new vertex and an edge connecting this vertex to an existing vertex of G. We give various sufficient conditions and necessary conditions on a subset S of the vertices of G so that the graph G∪W(S), obtained from G by adding a whisker to each vertex in S, is a sequentially Cohen-Macaulay graph. For instance, we show that if S is a vertex cover of G, then G∪W(S) is a sequentially Cohen-Macaulay graph. On the other hand, we show that if G?S is not sequentially Cohen-Macaulay, then G∪W(S) is not a sequentially Cohen-Macaulay graph. Our work is inspired by and generalizes a result of Villarreal on the use of whiskers to get Cohen-Macaulay graphs.     

Equivalences and the complete hierarchy of intersection graphs of paths in a tree

An (h,s,t)-representation of a graph G consists of a collection of subtrees of a tree T, where each subtree corresponds to a vertex in G, such that (i) the maximum degree of T is at most h, (ii) every subtree has maximum degree at most s, (iii) there is an edge between two vertices in the graph G if and only if the corresponding subtrees have at least t vertices in common in T. The class of graphs that have an (h,s,t)-representation is denoted by [h,s,t]. It is well known that the class of chordal graphs corresponds to the class [3, 3, 1]. Moreover, it was proved by Jamison and Mulder that chordal graphs correspond to orthodox-[3, 3, 1] graphs defined below.In this paper, we investigate the class of [h,2,t] graphs, i.e., the intersection graphs of paths in a tree. The [h,2,1] graphs are also known as path graphs [F. Gavril, A recognition algorithm for the intersection graphs of paths in trees, Discrete Math. 23 (1978) 211-227] or VPT graphs [M.C. Golumbic, R.E. Jamison, Edge and vertex intersection of paths in a tree, Discrete Math. 55 (1985) 151-159], and [h,2,2] graphs are known as the EPT graphs. We consider variations of [h,2,t] by three main parameters: h, t and whether the graph has an orthodox representation. We give the complete hierarchy of relationships between the classes of weakly chordal, chordal, [h,2,t] and orthodox-[h,2,t] graphs for varied values of h and t.     

Graphs with 1-Hamiltonian-connected cubes

The cube G³ of a connected graph G is that graph having the same vertex set as G and in which two distinct vertices are adjacent if and only if their distance in G is at most three. A Hamiltonian-connected graph has the property that every two distinct vertices are joined by a Hamiltonian path. A graph G is 1-Hamiltonian-connected if, for every vertex w of G, the graphs G and G?w are Hamiltonian-connected. A characterization of graphs whose cubes are 1-Hamiltonian-connected is presented.     

The strong metric dimension of graphs and digraphs

Let G be a connected (di)graph. A vertex w is said to strongly resolve a pair u,v of vertices of G if there exists some shortest u-w path containing v or some shortest v-w path containing u. A set W of vertices is a strong resolving set for G if every pair of vertices of G is strongly resolved by some vertex of W. The smallest cardinality of a strong resolving set for G is called the strong dimension of G. It is shown that the problem of finding the strong dimension of a connected graph can be transformed to the problem of finding the vertex covering number of a graph. Moreover, it is shown that computing this invariant is NP-hard. Related invariants for directed graphs are defined and studied.     

On the geodetic number of median graphs

A set of vertices S in a graph is called geodetic if every vertex of this graph lies on some shortest path between two vertices from S. In this paper, minimum geodetic sets in median graphs are studied with respect to the operation of peripheral expansion. Along the way geodetic sets of median prisms are considered and median graphs that possess a geodetic set of size two are characterized.     

Two spectral characterizations of regular, bipartite graphs with five eigenvalues

Graphs with a few distinct eigenvalues usually possess an interesting combinatorial structure. We show that regular, bipartite graphs with at most six distinct eigenvalues have the property that each vertex belongs to the constant number of quadrangles. This enables to determine, from the spectrum alone, the feasible families of numbers of common neighbors for each vertex with other vertices in its part. For particular spectra, such as [6,2⁹,0⁶,-2⁹,-6] (where exponents denote eigenvalue multiplicities), there is a unique such family, which makes it possible to characterize all graphs with this spectrum.Using this lemma we also to show that, for r?2, a graph has spectrum if and only if it is a graph of a 1-resolvable transversal design TD(r,r), i.e., if it corresponds to the complete set of mutually orthogonal Latin squares of size r in a well-defined manner.     

On the reduced signless Laplacian spectrum of a degree maximal graph

For a (simple) graph G, the signless Laplacian of G is the matrix A(G)+D(G), where A(G) is the adjacency matrix and D(G) is the diagonal matrix of vertex degrees of G; the reduced signless Laplacian of G is the matrix Δ(G)+B(G), where B(G) is the reduced adjacency matrix of G and Δ(G) is the diagonal matrix whose diagonal entries are the common degrees for vertices belonging to the same neighborhood equivalence class of G. A graph is said to be (degree) maximal if it is connected and its degree sequence is not majorized by the degree sequence of any other connected graph. For a maximal graph, we obtain a formula for the characteristic polynomial of its reduced signless Laplacian and use the formula to derive a localization result for its reduced signless Laplacian eigenvalues, and to compare the signless Laplacian spectral radii of two well-known maximal graphs. We also obtain a necessary condition for a maximal graph to have maximal signless Laplacian spectral radius among all connected graphs with given numbers of vertices and edges.     

3‐Factor‐Criticality of Vertex‐Transitive Graphs

A graph of order n is p ‐factor‐critical, where p is an integer of the same parity as n, if the removal of any set of p vertices results in a graph with a perfect matching. 1‐factor‐critical graphs and 2‐factor‐critical graphs are factor‐critical graphs and bicritical graphs, respectively. It is well known that every connected vertex‐transitive graph of odd order is factor‐critical and every connected nonbipartite vertex‐transitive graph of even order is bicritical. In this article, we show that a simple connected vertex‐transitive graph of odd order at least five is 3‐factor‐critical if and only if it is not a cycle.