A lower bound on the average size of a connected vertex set of a graph

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Abstract

The topic is the average order of a connected induced subgraph of a graph. This generalizes, to graphs in general, the average order of a subtree of a tree. In 1983, Jamison proved that the average order of a subtree, over all trees of order n, is minimized by the path Pn. In 2018, Kroeker, Mol, and Oellermann conjectured that Pn minimizes the average order of a connected induced subgraph over all connected graphs. The main result of this paper confirms this conjecture.

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 UV. 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 C=C(G). The number of connected sets in G will be denoted by N(G). LetS(G)=UC|U| be the sum of the sizes of the connected sets. Further, let n denote the order of G andA(G)=S(G)N(G)andD(G)=A(G)n denote, respectively, the average size of a connected set of G and the proportion of vertices in an average size connected set. The parameter D(G) 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 Kn, then A(Kn) is the average size of a nonempty subset of an n-element set, which is n2n12n1, the density then being 2n12n1, which is asymptotically 1/2.

There are a number of papers on the average size and density of connected sets in trees. The invariant A(G), 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 Pn. In particular A(T)(n+2)/3 for every tree T of order n with equality only for Pn; therefore D(T)>1/3 for every tree. Kroeker, Mol, and Oellermann conjectured in their 2018 paper [7] that Pn 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, thenA(G)n+23, with equality if and only if G is a path. In particular, D(G)>1/3 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 D(Tn) of a sequence Tn of trees tends to 1, then the proportion of vertices of degree 2 in Tn 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 N(G,H),S(G,H), and A(G,H) 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 H={x} is a singleton, then we write N(G,x),S(G,x) and A(G,x), respectively. Jamison [5, Theorem 4.6] proved the statement of the following theorem for trees.

Theorem 3.1

If HV is a connected subset of size h1 of

Near trees

It will be helpful in investigating graphs with at least one cut-vertex to consider the block-cut tree T=T(G) of a graph G. The vertex set of T 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 T if x lies in B. Call a block B of G a leaf if it is a leaf of the tree T(G); otherwise call B interior. Color the vertices in T 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 Gx denote the subgraph of G induced by V(G){x}. Let T be a shortest-path spanning tree of G rooted at x. For each connected set UC(Gx) of vertices in Gx, fix a vertex vU that is closest to x, with distance being the length of the path pU in T between vU and x. Let U=UpU, where we regard a path as its set of vertices. Let C(G,x) denote the set of connected sets in G containing vertex x. For QC(G,x), letW(Q)={U:UC(G

An inequality for graphs with a cut-vertex

Let x be a cut-vertex of a connected graph G of order n, and let M=M(x)=M(G,x) denote the number of connected components of Gx. Denote these components by G1,,GM, and let n1,,nM be their respective orders. Note that n=1+n1+n2++nM. For i=1,2,,M, denote by Gi the subgraph of G induced by the vertices V(Gi){x}. To simplify notation, let Ni=N(Gi) and Ni(x)=N(Gi,x). Let ai=av(Gi,x)/ni. Note that ai1 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 {x1,,xn}, then S(G)=i=1nN(G,xi).

Proof

Count the number of pairs (x,U) such that xV(G),UC(G) and xU, in two ways to obtainS(G)=UC|U|=xV(G)N(G,x)=i=1nN(G,xi). 

Theorem 7.2

For a connected graph G of order n we haveA(G)n+23, with equality if and only if G is a path.

Proof

The proof is by induction on n. The statement is easily checked for n4. By Lemma 4.3, it is also true for near trees as in Definition 4.2. Assume it is true for graphs of order n1, 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 D(G)>1/2 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 D(G)>12.

One difficulty in proving this conjecture, if true, is that knowing exactly for which graphs D(G)>12

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This work was partially supported by a grant from the Simons Foundation (322515 to Andrew Vince).

I would like to thank the referees for a thorough reading. Their comments substantially improved the quality of the paper.

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