A lower bound on the average size of a connected vertex set of a graph☆
Introduction
Although connectivity is a basic concept in graph theory, problems involving the enumeration of the connected induced subgraphs of a graph have only recently received attention. The topic of this paper is the average order of a connected induced subgraph of a graph. Let G be a connected finite simple graph with vertex set V, and let . The set U is said to be a connected set if the subgraph of G induced by U is connected. Denote the collection of all connected sets, excluding the empty set, by . The number of connected sets in G will be denoted by . Let be the sum of the sizes of the connected sets. Further, let n denote the order of G and denote, respectively, the average size of a connected set of G and the proportion of vertices in an average size connected set. The parameter is referred to as the density of connected sets of vertices. The density allows us to compare the average size of connected sets of graphs of different orders. The density is also the probability that a vertex chosen at random from G will belong to a randomly chosen connected set of G. If, for example, G is the complete graph , then is the average size of a nonempty subset of an n-element set, which is , the density then being , which is asymptotically 1/2.
There are a number of papers on the average size and density of connected sets in trees. The invariant , in this case, is the average order of a subtree of a tree. Although results are known for trees, beginning with Jamison's 1983 paper [5], nearly nothing is known for graphs in general. We review the literature in Section 2. Concerning lower bounds, Jamison proved that the density, over all trees of order n, is minimized by the path . In particular for every tree T of order n with equality only for ; therefore for every tree. Kroeker, Mol, and Oellermann conjectured in their 2018 paper [7] that minimizes the average size of a connected set over all connected graphs. The main result of this paper confirms this conjecture.
Theorem 1.1 If G is a connected graph of order n, then with equality if and only if G is a path. In particular, for all connected graphs G.
After reviewing the relevant literature in Section 2, each of the Sections 3, 4, 5 and 6 contain a preliminary result required for the proof of Theorem 1.1. In Section 3, the result (Theorem 3.1) concerns the average size a connected set of G containing a fixed connected subset H. In Section 4, the result (Lemma 4.3) is that certain very sparse graphs satisfy the inequality in Theorem 1.1. In Section 5, the result (Theorem 5.1) gives an inequality relating the number of connected sets containing a given vertex x to the number of connected sets not containing x. In Section 6, the result (Theorem 6.1) is an essential inequality valid for graphs with at least one cut-vertex. Section 7 provides the final step in the proof of Theorem 1.1. Two problems that remain open are discussed in Section 8.
Section snippets
Previous results
Following Jamison's study [5], a number of papers on the average order of a subtree of a tree followed [3], [6], [8], [9], [10], [12], [13]. Concerning upper bounds, Jamison [5] provided a sequence of trees (certain “batons”) showing that there are trees with density arbitrarily close to 1. However, if the density of a sequence of trees tends to 1, then the proportion of vertices of degree 2 in must also tend to 1. This led to the question of upper and lower bounds on the density
The average size of connected sets containing a given connected set
If V is the set of vertices of a connected graph G and H is a connected subset of V, let , and denote the number of connected sets in G containing H, the sum of the sizes of all connected sets containing H, and the average size of a connected set containing H, respectively. If is a singleton, then we write and , respectively. Jamison [5, Theorem 4.6] proved the statement of the following theorem for trees.
Theorem 3.1 If is a connected subset of size of
Near trees
It will be helpful in investigating graphs with at least one cut-vertex to consider the block-cut tree of a graph G. The vertex set of is the union of the cut-vertices of G and the blocks, i.e., the maximal 2-connected components, of G. The latter includes the cut-edges of G. A cut-vertex x and a block B are adjacent in if x lies in B. Call a block B of G a leaf if it is a leaf of the tree ; otherwise call B interior. Color the vertices in corresponding to a block B in G red if
An inequality relating the number of connected sets containing a given vertex to the number of connected sets not containing the vertex
Let G be a connected graph and x a vertex of G. For ease of notation, let denote the subgraph of G induced by . Let T be a shortest-path spanning tree of G rooted at x. For each connected set of vertices in , fix a vertex that is closest to x, with distance being the length of the path in T between and x. Let , where we regard a path as its set of vertices. Let denote the set of connected sets in G containing vertex x. For , let
An inequality for graphs with a cut-vertex
Let x be a cut-vertex of a connected graph G of order n, and let denote the number of connected components of . Denote these components by , and let be their respective orders. Note that . For , denote by the subgraph of G induced by the vertices . To simplify notation, let and . Let . Note that for all i.
The main result of this section is the following inequality, which is essential
Proof of the lower bound theorem
Proposition 7.1 If G is a connected graph with vertex set , then .
Proof Count the number of pairs such that and , in two ways to obtain □
Theorem 7.2 For a connected graph G of order n we have with equality if and only if G is a path.
Proof The proof is by induction on n. The statement is easily checked for . By Lemma 4.3, it is also true for near trees as in Definition 4.2. Assume it is true for graphs of order , and let G have
Two open problems
Although, for a general connected graph, the lower bound of Theorem 1.1 is best possible, evidence indicates that for a large class of graphs. The result of Kroeker, Mol, and Oellermann [7] referenced in Section 2, for example, proves that this is the case for cographs. We made the following conjecture in [11].
Conjecture 1 For any graph G, all of whose vertices have degree at least 3, we have .
One difficulty in proving this conjecture, if true, is that knowing exactly for which graphs
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