The average size of independent sets in unicyclic graphs

The average size of independent sets of a graph ( avi ) is considered. It can be viewed as the logarithmic derivative of the independence polynomial at 1 . Lower and upper bounds on avi for unicyclic graphs of a given order are determined, and the respective extremal graphs are given. The unicyclic graphs that maximize (minimize) avi coincide with those that maximize (minimize, respectively) the number of independent sets.


Introduction
In 1980s, Jamison [21,22] initiated the study of the average size of subtrees of a tree. He studied an extremal problem and proved that the path P n has the smallest average size of subtrees among all trees of a fixed order n. However, although the average size of subtrees of a tree has been extensively studied (see [8,25,27,31,34]), the problem of describing the tree(s) of a given order with the largest average size of subtrees remains unresolved. Inspired by these works, the study of the average size of certain substructures of graphs, including subtrees [9,10], independent sets [1], matching [2], dominating sets [5,14] and connected sets [15,30], has attracted increasing attention in recent years.
An independent set of a graph G is a set of pairwise non-adjacent vertices of G. The total number of such sets has been studied extensively and is usually called Merrifield-Simmons index in honor of the work of Merrifield and Simmons in mathematical chemistry [26]. In fact it has been called the Fibonacci number of G (cf. [23]), based on the fact that in the case of a path, i(P n ) = F n+2 , where F h is the standard h-th Fibonacci number, F 1 = 1 , , F 2 = 1 , F 3 = 2 , F h+2 = F h+1 + F h . In addition, the association of i(G) with the hard core model is well established in statistical physics [7]. It is of interest to determine the graphs having extremal (maximal or minimal) i(G)-values in several given graph classes, especially in the context of mathematical chemistry, such as trees, unicyclic graphs, bicyclic graphs and so on (see [13,29,35,36,39,40]). For more extremal results on i(G) see the surveys [32,33] and for related results see [19,20,37,38].
This paper is focused on the extremal problems of the average size of independent sets (avi), instead of their number (i(G)). Different authors have already studied bounds on this invariant for specific graphs, see [11,12,24]. Recently Andriantiana, Misanantenaina, and Wagner [1] studied the extreme problem of this invariant and showed that the complete and empty graphs attain the minimum and maximum among all graphs of a given order, whereas the path and the star attain the minimum and maximum among all trees of a given order. It turns out that these results parallel the extremal graph for the number of independent sets. In view of the definition of correlation, this may not be too surprising. However, the correlation is not fully understood between these two invariants (the average size of independent sets and the number of independent sets).
A natural question is whether the extremal graph for the average size of independent sets in other graph classes remains parallel to the extremal graph for the number of independent sets. A unicyclic graph, as a natural extension of a tree, is a connected graph containing exactly one cycle. Many extremal problems, considered for various classes of trees, can be extended to unicyclic graphs. For example, Pedersen and Vestergaard [29] determined sharp lower and upper bounds for the Merrifield-Simmons index in a unicyclic graph of a given order. Andriantiana and Wang [4] characterized the extremal unicyclic graphs with respect to the number of subtrees. Ou [28] described the unicyclic graphs of order n with the largest and second largest Hosoya index. Hou [17] determined the unicyclic graphs of order n with the minimal energy. Hou, Gutman, and Woo [18] characterized the unicyclic bipartite graphs of order n having the maximal energy. Andriantiana and Wagner [3] also found the non-bipartite unicyclic graphs of order n with the maximum energy.
In this paper, we study the average size of independent sets in a unicyclic graph and characterized the unicyclic graphs of order n having the largest and smallest average size of independent sets. These turn out that the unicyclic graphs that maximize (minimize) the average size of independent sets are those that also maximize (minimize, respectively) the number of independent sets. Section 2 gives some auxiliary results that will be used for the subsequent proofs. In Section 3, we determine the graphs minimizing and maximizing the average size of independent sets among unicyclic graphs with the given order. We conclude in Section 4 with some questions.

Preliminaries
Let G be a simple graph with vertex set V (G) and edge set E(G). The number of vertices of G is called the order of G, , and a vertex of degree 1 is called a leaf vertex. For a vertex subset X ⊆ V (G), we write G − X for the graph obtained from G by removing all the vertices in X and the edges incident with them. The distance d(u, v) between two vertices u and v of G is the number of edges on a shortest path from u to v in G. Let v be a vertex of G and S ⊆ V (G), the distance between v and S is d As usual, we use P n , S n , and C n to denote the path, star, and cycle, respectively, of order n. For n ≥ 3, we denote by U S n the graph obtained by inserting an edge connecting two leaves of S n , and U P n the graph obtained by identifying one end of P n−2 with one of the vertices of C 3 (see Figure 1). For all other notations we refer to [6]. For a graph G, an independent set in G is a set of vertices with no two adjacent vertices. Let i(G, k) be the number of independent sets of cardinality k in G. It is both consistent and convenient to set i(G, 0) = 1. The independence polynomial of G is defined by One can easily see that I(G, 1) = k≥0 i(G, k) is the total number of independent sets of G, and that I (G, 1) = k≥0 ki(G, k) is the total size of all independent sets of G. For brevity we use the notations i(G) = I(G, 1) and t(G) = I (G, 1). Then the average size of independent sets in G is given by the logarithmic derivative The following results are very useful for calculating the independence polynomial.
Proposition 2.1 (see [16]). Let v be a vertex of G and G 1 , G 2 , . . . , G k be the disjoint components of G. Then A recursive formula for i(G) and t(G) is obtained as a immediate consequence.
Proposition 2.2 (see [1]). For any graph G and v ∈ V (G), we have For the equality in Proposition 2.1 (ii), by taking the logarithm and the differentiation on both sides, letting x = 1, we get the next result. Proposition 2.3. Let G 1 , G 2 , . . . , G k be the disjoint components of G. Then The following result established in [1], will be fundamental for proving our main results. Lemma 2.1 (see [1]). For every tree T of order n, we have avi(S n ) ≥ avi(T ) ≥ avi(P n ). Moreover, > avi(P n ).

Main results
For notational simplicity, in this section we always set It is known that U S n maximizes the number of independent sets and C n and U P n minimizes the number of independent sets among all unicyclic graphs of order n (see [29]). We show that these results parallel the extremal graph for the average size of independent sets. We begin with some lemmas that are useful for proving the subsequent results. Moreover, for n ≥ 5 with equality only for n = 6. For n ≥ 7, with equality only for n = 8.
Let u be one of the vertices of degree 2 that lies on the unique cycle of U P n and v be any vertex of C n . By applying Proposition 2.1, we have [23]). Moreover, by Proposition 2.2, we have the recursion t (P n ) = t (P n−2 ) + i (P n−2 ) + t (P n−1 ) with t (P 1 ) = 1 and t (P 2 ) = 2. By solving this recursive equation we get On the other hand, it follows from Proposition 2.
We prove that the absolute value of c n is decreasing. When n ≥ 5, we get Therefore, |c n | is decreasing for n. Meanwhile, observe that the sign of c n alternates, i.e., c n < 0 if n is odd and c n > 0 if n is even. This implies that avi(C n ) is alternately greater and less than an. It follows that if n ≥ 5, the maximum of c n is attained for n = 6, and if n ≥ 7, the maximum of c n is attained for n = 8. In both cases, the values of avi(C n ) are easy to calculate and the two inequalities in Lemma 3.1 follow.

Lemma 3.2.
For any graph G and v ∈ V (G), we have This establishes the left inequality. And the right inequality follows easily from the fact that G − v is a proper subgraph of G. (ii) for any two vertices u, v of T , we have avi(T − u − v) ≥ a(n − 2) + c for n ≥ 3.
Proof. We only prove (i), whereas the proofs of (ii) is similar and is omitted. We first characterize the graphs minimizing the average size of independent sets among all unicyclic graphs of order n.
with equality if and only if G ∈ {C n , U P n }.
Proof. If G ∈ {C n , U P n }, then by Lemma 3.1, Assume that G / ∈ {C n , U P n } is a unicyclic graph of order n ≥ 3. It suffices to prove that avi (G) > avi (C n ). We write C to denote the unique cycle in G and consider a vertex Assume that k ≥ 5 and let T = G − T 1 . Repeating the above calculation yields For simplicity, we use σ and σ to denote i(G−v) i(G) and i(T ) , respectively, and set Note that and likewise Applying Lemma 3.2, we deduce that Combining (4), (5) and (6), we get σ > σ . Now, we write Observe that T is not a path. Using Lemma 2.1 (iii) gives avi (T ) > a|T |+d. Moreover, by Lemma 2.
If |T 1 | = 2, then T 1 ∼ = P 2 . Thus we have avi ( By applying Equality (4) and Lemma 3.2, we obtain Observe that in this case n ≥ 7. Therefore by Lemma 3.1, This completes the proof of k ≥ 5.
Next we deal with the cases of k = 4 and k = 3. In each case, we divide into two subcases |T 1 | ≥ 3 and |T 1 | = 2. Then for each subcase we continue to distinguish different cases that depend on how many T i s are a single vertex for i ≥ 2. This yields eight cases for k = 4 and six cases for k = 3. Now, we return to the expression For each of these 14 cases, we employ the following bounds and values, which can be easily obtained by Theorems 2.1 and 3.3 or simple calculations.
Also by Theorem 3.2 and Inequality (6), we have We plug these above bounds and values into (8), and use the expression (4) to obtain the maximum or minimum possible value of σ. Since the expression (8) is linear in σ, its minimum is either obtained at the maximum or minimum value of σ. Then we get lower bounds of avi(G) for the above 14 cases, which are easily calculated by computer. We consider, as an example, the case with the worst bound, which is obtained for |T 1 | ≥ 3, |T 2 | > 1, |T 3 | > 1, |T 4 | > 1 when k = 4. Then Note that n ≥ 7 in this case, combining with the Lemma 3.1, we have In order to illustrate a more general process, as another example, we consider the case with the second-worst bound, which is obtained for |T 1 | = 2, |T 2 | > 1, |T 3 | > 1 when k = 3. Then Note that n ≥ 5 in this case. By Lemma 3.1, we have The other 12 cases can be treated analogously. Each time we obtain avi(G) ≥ an + ε where ε > 0.0104 is a constant. Note that we have n ≥ 5 for all these 12 cases. Therefore for each of these we have avi(G) ≥ an + ε > avi(C n ) by Lemma 3.1. Now, only the case k = 2 remains. Consider the following subcases.
Assume that G / ∈ {C n , U S n } is a unicyclic graph. It suffices to prove that avi(G) < avi(U S n ). Then we proceed by induction on n (note that n ≥ 5 since G / ∈ {C n , U S n }). For n = 5, we easily see that G ∼ = U P 5 , and thus by Theorem 3.1, avi(G) < avi(U S n ) . We write C for the unique cycle of G and let v ∈ V (G) that has the maximum distance from C. Clearly, v is a leaf. Assume that vu ∈ E(G) with d G (u) = k. We first prove the following inequality: Consider the distance between v and C.
consists of a tree of order n − k + 1, say T , and k − 3 singletons. Moreover, since G / ∈ {C n , U S n }, we must have n − k ≥ 2. By Proposition 2.3, Lemma 2.1 and Equality (1), deduce that consists of a unicyclic graph of order n − k, say G , and k − 2 singletons, where n − k ≥ 3. By Proposition 2.3, induction hypothesis and Equality (2), we have Therefore, the inequality (9) is proved.

Concluding remarks
Considering the family of unicyclic graphs of order n ≥ 3, we have proved that the minimum average size of independent sets is attained by C n and U P n , while the graph U S n resulting from adding one edge between two leaves of the star of order n uniquely realizes the maximum. These results parallel the extremal graphs for the number of independent sets. This leaves some natural questions. First, we can consider the average size of independent sets by choosing any different graph instead of unicyclic graph, and see if there are similar parallel conclusions. Secondly, it is well-known that there is a negative correlation between Merrifield-Simmons index and Hosoya index, which are the number of independent sets and the number of matchings, respectively. Recently, Andriantiana et al. [2] also considered the extreme problem on the average size of matching of graphs. It was shown that in all the instances they dealt with, including general graphs and trees of given order, graphs that minimize the average size of the independent set also maximize the average size of the matching, and vice versa. So, it is also interesting to consider whether the average size of independent set and the average size of matching also have a negative correlation in other graph classes.