Average distance in graphs and eigenvalues

Brendan McKay gave the following formula relating the average distance between pairs of vertices in a tree T and the eigenvalues of its Laplacian:

     

On packing bipartite graphs

G andH, two simple graphs, can be packed ifG is isomorphic to a subgraph of , the complement ofH. A theorem of Catlin, Spencer and Sauer gives a sufficient condition for the existence of packing in terms of the product of the maximal degrees ofG andH. We improve this theorem for bipartite graphs. Our condition involves products of a maximum degree with an average degree. Our relaxed condition still guarantees a packing of the two bipartite graphs.the paper was written while the authors were graduate students at the University of Chicago and was completed while the first author was at M.I.T. The work of the first author was supported in part by the Air Force under Contract OSR-86-0076 and by DIMACS (Center for Discrete Mathematics and Theoretical Computer Science), a National Science Foundation Science and Technology Center-NSF-STC88-09648. The work of the second author was supported in part by NSF grant CCR-8706518.     

Integral distance graphs

Suppose D is a subset of all positive integers. The distance graph G(Z, D) with distance set D is the graph with vertex set Z, and two vertices x and y are adjacent if and only if |x − y| ≡ D. This paper studies the chromatic number χ(Z, D) of G(Z, D). In particular, we prove that χ(Z, D) ≤ |D| + 1 when |D| is finite. Exact values of χ(G, D) are also determined for some D with |D| = 3. © 1997 John Wiley & Sons, Inc. J Graph Theory 25: 287–294, 1997     

Orientation distance graphs

For two nonisomorphic orientations D and D′ of a graph G, the orientation distance d_o(D,D′) between D and D′ is the minimum number of arcs of D whose directions must be reversed to produce an orientation isomorphic to D′. The orientation distance graph 𝒟_o(G) of G has the set 𝒪(G) of pairwise nonisomorphic orientations of G as its vertex set and two vertices D and D′ of 𝒟₀(G) are adjacent if and only if d_o(D,D′) = 1. For a nonempty subset S of 𝒪(G), the orientation distance graph 𝒟₀(S) of S is the induced subgraph 〈S〉 of 𝒟_o(G). A graph H is an orientation distance graph if there exists a graph G and a set S⊆ 𝒪(G) such that 𝒟_o(S) is isomorphic to H. In this case, H is said to be an orientation distance graph with respect to G. This paper deals primarily with orientation distance graphs with respect to paths. For every integer n ≥ 4, it is shown that 𝒟_o(P_n) is Hamiltonian if and only if n is even. Also, the orientation distance graph of a path of odd order is bipartite. Furthermore, every tree is an orientation distance graph with respect to some path, as is every cycle, and for n ≥ 3 the clique number of 𝒟_o(P_n) is 2 if n is odd and is 3 otherwise. © 2001 John Wiley & Sons, Inc. J Graph Theory 36: 230–241, 2001     

Average distance and maximum induced forest

With the help of the Graffiti system, Fajtlowicz conjectured around 1992 that the average distance between two vertices of a connected graph G is at most half the maximum order of an induced bipartite subgraph of G, denoted α₂(G). We prove a strengthening of this conjecture by showing that the average distance between two vertices of a connected graph G is at most half the maximum order of an induced forest, denoted F(G). Moreover, we characterize the graphs maximizing the average distance among all graphs G having a fixed number of vertices and a fixed value of F(G) or α₂(G). Finally, we conjecture that the average distance between two vertices of a connected graph is at most half the maximum order of an induced linear forest (where a linear forest is a union of paths). © 2008 Wiley Periodicals, Inc. J Graph Theory 60: 31–54, 2009     

Average distances and distance domination numbers

Let k be a positive integer and G be a simple connected graph with order n. The average distance μ(G) of G is defined to be the average value of distances over all pairs of vertices of G. A subset D of vertices in G is said to be a k-dominating set of G if every vertex of V(G)−D is within distance k from some vertex of D. The minimum cardinality among all k-dominating sets of G is called the k-domination number γ_k(G) of G. In this paper tight upper bounds are established for μ(G), as functions of n, k and γ_k(G), which generalizes the earlier results of Dankelmann [P. Dankelmann, Average distance and domination number, Discrete Appl. Math. 80 (1997) 21-35] for k=1.     

Average distance and domination number revisited

We give a short proof for Dankelmann’s characterization of connected graphs of given order and domination number that maximize the average distance [P. Dankelmann, Average distance and the domination number, Discrete Appl. Math. 80 (1997) 21-35].     

Average eccentricity,k-packing and k-domination in graphs

Let $G$ be a connected graph. The eccentricity $e\left(v\right)$ of a vertex $v$ is the distance from $v$ to a vertex farthest from $v$. The average eccentricity $\mathrm{avec}\left(G\right)$ of $G$ is defined as the average of the eccentricities of the vertices of $G$, i.e., as $\frac{1}{\left|V\right|}{\sum }_{v\in V}e\left(v\right)$, where $V$ is the vertex set of $G$. For $k\in \mathbb{N}$, the $k$-packing number of $G$ is the largest cardinality of a set of vertices of $G$ whose pairwise distance is greater than $k$. A $k$-dominating set of $G$ is a set $S$ of vertices such that every vertex of $G$ is within distance $k$ from some vertex of $S$. The $k$-domination number (connected $k$-domination number) of $G$ is the minimum cardinality of a $k$-dominating set (of a $k$-dominating set that induces a connected subgraph) of $G$. For $k=1$, the $k$-packing number, the $k$-domination number and the connected $k$-domination number are the independence number, the domination number and the connected domination number, respectively. In this paper we present upper bounds on the average eccentricity of graphs in terms of order and either $k$-packing number, $k$-domination number or connected $k$-domination number.     

Colouring prime distance graphs

Four colours are necessary and sufficient to colour all the integers so that any two with difference equal to a prime have different colours. We investigate the corresponding problem when the setD of prescribed differences is a proper subset of the primes. In particular, we prove that ifD contains {2, 3} and also contains a pair of twin primes (one of which may be 3), then four colours are necessary. Numerous results regarding periodic colourings are also obtained. However, the problem of characterizing those setsD which necessitate four colours remains open.     

Boolean distance for graphs

The boolean distance between two points x and y of a connected graph G is defined as the set of all points on all paths joining x and y in G (Ø if x = y). It is determined in terms of the block-cutpoint graph of G, and shown to satisfy the triangle inequality b(x,y)? b(x, z)∪b(z,y). We denote by B(G) the collection of distinct boolean distances of G and by M(G) the multiset of the distances together with the number of occurrences of each of them. Then $\text{B(G)}\text{= 1+(}{}^{\mathrm{b+1}}{}_2\text{)}$ where b is the number of blocks of G. A combinatorial characterization is given for B(T) where T is a tree. Finally, G is reconstructible from M(G) if and only if every block of G is a line or a triangle.     

Long cycles and paths in distance graphs

For n∈N and D⊆N, the distance graph has vertex set {0,1,…,n−1} and edge set {ij∣0≤i,j≤n−1,|j−i|∈D}. Note that the important and very well-studied circulant graphs coincide with the regular distance graphs.A fundamental result concerning circulant graphs is that for these graphs, a simple greatest common divisor condition, their connectivity, and the existence of a Hamiltonian cycle are all equivalent. Our main result suitably extends this equivalence to distance graphs. We prove that for a finite set D of order at least 2, there is a constant c_D such that the greatest common divisor of the integers in D is 1 if and only if for every n, has a component of order at least n−c_D if and only if for every n≥c_D+3, has a cycle of order at least n−c_D. Furthermore, we discuss some consequences and variants of this result.     

Co-universal C*-algebras associated to generalised graphs

We introduce P-graphs, which are generalisations of directed graphs in which paths have a degree in a semigroup P rather than a length in ?. We focus on semigroups P arising as part of a quasi-lattice ordered group (G, P) in the sense of Nica, and on P-graphs which are finitely aligned in the sense of Raeburn and Sims. We show that each finitely aligned P-graph admits a C*-algebra C*_min (Λ) which is co-universal for partialisometric representations of Λ which admit a coaction of G compatible with the P-valued length function. We also characterise when a homomorphism induced by the co-universal property is injective. Our results combined with those of Spielberg show that every Kirchberg algebra is Morita equivalent to C*_min (Λ) for some (?²* ?)-graph Λ.