Introduction

Evidently, a hypertext forms a network of documents mostly linked on the basis of content-related connections. There is a range of studies applying the compactness measure itroduced in [2] in order to answer questions concerning the structure of hypermedia [1–8]. All these studies addressed compactness computing the values of particular graph invariants which implies high computational costs. In this paper, we take a different perspective considering the limit of compactness for different classes of connected graphs proof-theoretically. Our approach allows for omitting the computational step when the conditions below hold.

The paper is organized as follows. Section starts with repeating graph-theoretical notions used throughout the paper. Section outlines our main findings regarding the limit value of compactness. Section illustrates the application of our tool on four selected graph classes. Finally, Section summarizes our mathematical findings and gives an outlook on results obtained which are part of a subsequent publication. More specifically, in Section we give an overview of those graph classes for which compactness can be easily obtained applying our mathemaical tool (in fact, we have studied about 30 well-known graph classes, there are presumably more than those mentioned here for which our tool can be applied).

Preliminaries

In this Section, we recall some definitions from graph theory to be used throughout this paper. Let G be a simple undirected graph with the vertex set V = V(G) and the edge set E = E(G). The order n of G is the number of its vertices (n = |V|). The size of G is the number of its edges.

Definition 1. The degree (or valency) deg(v) of a vertex v of a graph G is the number of edges incident to v in G.

Definition 2. The geodesic distance δ(v, w) of two vertices u and v in graph G is the number of edges of the shortest path in G connecting them.

Definition 3. The diameter D(G) of a graph G is the maximum of geodesic distances in G.

By L(G) we denote the average geodesic distance in graph G = (V, E) [9]: (1)

Further, we denote the numerator of the fraction in (1) by Σ(G), that is: (2)

Thus, (1) can be rewritten as: (3)

Further, for every vertex c ∈ V we denote by Σ(c, G) the sum of n − 1 geodesic distances from c to vertices in V \ {c}. That is: (4) and using this notation we write (5)

For example, for the path graph P₂ on two vertices u and v connected by an edge we get and

We repeat the definition of the compactness C_B(G) of a graph G = (V, E), |V| = n > 1, as introduced in [2] in a version obtained from [1]: (6) where K is “the maximum value an entry in the converted distance matrix [of a graph] can assume” [2, p. 161], Com(G) is the set of connected components of G and |G′| is the order of the graph G′ (connected component of the graph G). In what follows, we set K = n. Here, we consider only connected graphs, so we can obviously write the following: (7)

On can easily see that C_B(G) ≤ 1. Further, since ∀{v, w} ∈ [V]²: D(G) ≥ δ(v, w), we have: (8)

Thus, with (7) and (8) we get for every connected graph G: (9)

Definition 4. The path graph P_m, m ≥ 2, is a simple connected undirected graph with two vertices of degree 1 (called terminal vertices) and m − 2 vertices of degree 2 (called internal vertices).

The order n of P_m is equal to m and its diameter D(P_m) = m − 1. The vertices of P_m can be labeled by the consecutive integers {1, 2, …, m} in such a way that the terminal vertices are labeled by 1 and m, respectively, and for every integer i, 1 ≤ i ≤ m − 1, the consecutive vertices with labels i and i + 1 are adjacent.

Further we need the following formula the proof of which one can easily get with the straightforward calculation: (10)

Hence in view of (3) we have and (11)

Definition 5. The Cartesian product G₁☐G₂ of two graphs G₁, G₂ is a graph with vertex set V(G₁) × V(G₂ ) such that any two vertices (v, u), (w, z) ∈ V(G₁☐G₂ ) are adjacent iff v = w and u and z are adjacent in G₂ or v and w are adjacent in G₁ and u = z.

Remark 1. If G = G₁☐G₂ is the Cartesian product of two graphs G₁ of order n₁ and G₂ of order n₂, then the following properties (referred to below) hold:

• G is connected iff both G₁ and G₂ are connected;
• the diameter of G is the sum of the diameters of G₁ and G₂:
• the order n of G is the product n₁ n₂ of the order n₁ of G₁ and the order n₂ of G₂.

Example 1. Let us consider the Cartesian product G(m) of two copies of the path graph P_m, that is, G(m) = P_m☐P_m (a so-called square lattice graph whose compactness C_B is investigated below). According to Remark 1, G(m) is connected (since P_m is connected), its order is m² and its diameter D(G(m)) is 2(m − 1).

Main results

Throughout the present paper we deal with sequences {G(m)|m = 1, 2, …} of connected graphs that satisfy the following “natural” condition (12) where n is the order of the graph G(m).

Theorem 1. Let {G(m)|m = 1, 2, …} be a sequence of simple undirected connected graphs G(m) such that the order n = n(m) → ∞ for m → ∞. Assume that the following holds: (13) where D(G(m)) is the diameter of the graph G(m). Then, the compactness C_B(G(m)) tends to 1 for m → ∞.

Proof. In view of (9) we have which implies with our assumptions that

Theorem 2. Let {G(m)|m = 1, 2, …} be a sequence of simple undirected connected graphs G(m) such that the order n = n(m)→∞ for m → ∞. Then, L(G(m))/n →0 for m → ∞ (n → ∞) iff D(G(m))/n → 0 for m → ∞ (n → ∞).

Proof. In view of (8), we can easily see that we only need to prove that if D(G)/n ↛ 0 for m → ∞ then L(G)/n ↛ 0 for m → ∞.

Without loss of generality we assume that

Hence, if we take any number a, 0 < a < c, then for all sufficiently large numbers m we have which implies that there is a geodesic path (subgraph P_k(n)) in G of length k(n) where k(n) is the integer part of the number an. So we have an = k(n) + ε_n with ε_n (0 ≤ ε_n < 1) being the fractional part of the number an. Therefore, in view of Σ(G) > Σ(P_k(n)) and with (10) we have for all sufficiently large m

Thus, with k(n) = an − ε_n we get which implies in view of lim_m→∞ n = ∞ that for all sufficiently large numbers m we have

Hence, L(G)/n ↛ 0.

From these two theorems obviously follows:

Corollary 1. For any sequence of simple undirected connected graphs G(m) for which the order n = n(m) of G(m) tends to ∞ whenever m → ∞, C_B(G(m)) → 1 for m → ∞ iff for m → ∞ L(G(m))/n → 0 (D(G(m))/n → 0).

And what about the case of disconnected graphs? It turns out that Corollary 1 can be easily generalized with some consideration for the case of disconnected graph classes. That is, the following statement holds:

Theorem 3. Let {G(m)|m = 1, 2, …} be a sequence of simple undirected not necessarily connected graphs G(m) such that the order n = n(m) → ∞ for m → ∞. Then, the compactness C_B of G(m) tends to 1 iff both of the following equalities hold:

1. L(G(m))/n →0 (or D(G(m))/n → 0) for m → ∞
2. lim_m→∞ n₁/n = 1 (equivalently, lim_m→∞(n − n₁)/n = 0), where n₁ is the order of the largest connected component of G(m).

These results give an answer to the question in which case C_B(G(m)) tends to 1 for m → ∞ (n → ∞).

Some simple applications

In this section, we consider four simple classes of undirected connected graphs and examine their compactness C_B in the limit of their order (i.e., n → ∞). Sometimes, C_B is easily estimated as in the case of complete graphs. In most cases, however, it is difficult to calculate the exact value of C_B or to give a good estimation of it. Here, we refer to Corollary 1 in order to do this.

The examples of graphs considered here have the following properties. Their diameter D(G) is either constant or grows slower than its order n in such a way that D(G)/n tends to 0 whenever n tends to ∞.

Concluding remarks

We confined us here to providing only four simple examples of the graph classes, for which our tool can be easily applied. Actually, we have found more than 30 well-known graphs classes for which our tool is applicable. So, the compactness of these graphs tends to 1 whenever their order tends to ∞.

First, among these graph classes there are those whose diameter does not depend on the order n. These are book graphs, complete r-partite graphs, crown graphs, Hadamard graphs, Keller graphs, lating square graphs, Paley graphs, strongly regular graphs, Turán graphs, wheel graphs, windmill graphs and some others.

Next, we found some graph classes for each of which the estimation of its diameter as a function of the order n allows the application of our tool. Among those graph classes are the following: de Bruijn graphs, cube-connected cycles, Fibonacci cube graphs, folded cube graphs, Hamming graphs, Johnson graphs, king’s graphs, Kneser graphs, knight’s graphs, perfect undirected binary trees, self-complementary graphs, Ramanujan graphs and others.

Further, we have seen in Section that the complete graph K_m has the largest possible value of compactness which is 1. So we can say that the graph K_m is the most compact graph among all the graphs of the same order m. If we try now to get the limit value of compactness of the path graph P_m using our tool, we see that this is not possible because D(P_m)/m ↛ 0 for m → ∞. Indeed, but with (11) we easily get lim_m→∞ C_B(P(m)) = 2/3. We can prove (to appear) that the path graph P_m is the least compact among all the simple connected undirected graphs of the same order m. That is, for each such graph G of order m the following holds:

This finding defines the range of possible values of compactness for connected graphs. Hence, the limit value of compactness for any sequence of simple connected undirected graphs lies within the interval [2/3; 1]. Moreover, we can prove (to appear) that for any number α in the interval [2/3; 1] a graph family can be constructed for which the limit value is exactly α.

It is worth noting that in the case of not necessarily connected graphs the value of compactness lies within the interval [0, 1]. Our future work will consider, amongst others, an extended set of graph classes and the study of a range of invariants including weighted and unweighted ones.

Acknowledgments

This work has been funded by German Federal Ministry of Education (BMBF) in the framework of the research project Linguistic Networks: Text Technological Representation, Computational Linguistic Synthesis and Physical Modeling and by the Deutsche Forschungsgemeinschaft (DFG) in the framework of the research project EcoGest (https://scs.techfak.uni-bielefeld.de/ecogest/). Financial support by the BMBF and the DFG is gratefully acknowledged.