On symmetric division deg index of unicyclic graphs and bicyclic graphs with given matching number

: Nowadays, it is an important task to ﬁnd extremal values on any molecular descriptor with respect to di ﬀ erent graph parameters. In a molecular graph, the vertices represent the atoms and the edges represent the chemical bonds in the terms of graph theory. For one thing, the molecular graphs of some chemical compounds are unicyclic graphs or bicyclic graphs, such as benzene compounds, napthalene, cycloalkane, et al. For another, the symmetric division deg index is proven to be a potentially useful molecular descriptor in quantitative structure-property / activity relationships (QSPR / QSAR) studies recently. Therefore, we present the maximum symmetric division deg indices of unicyclic graphs and bicyclic graphs with given matching number. Furthermore, we identify the corresponding extremal graphs.


Introduction
As a numerical parameter of molecular structure, topological molecular descriptors play an important role in chemistry, pharmacology and materials science, etc. (see [1][2][3]). Symmetric division deg (S DD for short) index is one of the 148 discrete Adriatic indices that showed good predictive abilities on the testing sets provided by International Academy of Mathematical Chemistry (IAMC) [4].
This graph invariant has a good correlation with the total surface area of polychlorobiphenyls [4], and its extremal graphs which have a particularly elegant and simple structure are obtained with the help of MathChem [5]. Let us write the definition of S DD index again, that is where d G (u) denotes the degree of vertex u in G. Recently, Furtula et al. [6] found that S DD index is an applicable and viable topological index, whose predictive capability is better than that of some popular topological indices, such as the famous geometric-arithmetic index and the second Zagreb index. Gupta et al. [7] determined some upper and lower bounds of S DD index on some classes of graphs and characterized the corresponding extremal graphs. For other recent mathematical investigations, the readers can refer [8][9][10][11][12][13][14][15][16]. At present, studying the behavior of topological indices is an essential subject. S DD index has been studied extensively since it was proved to be an applicable and viable molecular descriptor in 2018. Furthermore, unicyclic graphs and bicyclic graphs are two kinds of important graphs in mathematical chemistry because they can be seen as the molecular graphs of some chemical compounds. There are many papers on topological indices of unicyclic graphs and bicyclic graphs. Recent results can be referred to [17][18][19] et al. So we study the extremal values of S DD indices on unicyclic graphs and bicyclic graphs with given matching number and find the corresponding extremal graphs. Our results may be used to detect the chemical compounds that may have desirable properties. Namely, if one can find some properties well-correlated with this descriptor (S DD index has a good correlation with the total surface area of polychlorobiphenyls), the extremal graphs should correspond to molecules with minimum or maximum value of that property.
We only deal with connected graphs without multiple edges and loops. We use G = (V(G), E(G)) to denote the graph with vertex set V(G) and edge set E(G). Let N G (x) be the set of all neighbours of x ∈ V(G) in G, and d G (x) = |N G (x)|. If d G (x) = 1, we call x is a pendant vertex, and denoted by PV(G) the set of all pendant vertices in G. We denote the distance between vertices u and v of G by d G (u, v). Let G − xy and G − x be the graph obtained from G by deleting the edge xy ∈ E(G) and the vertex x ∈ V(G), respectively. Similarly, G + uv is the graph obtained from G by adding an edge uv E(G), where u, v ∈ V(G). Unicyclic graphs U and bicyclic graphs B are connected graphs satisfying |E(U)| = |V(U)| and |E(B)| = |V(B)| + 1, respectively. As usual, let's denote the path, the cycle and the star on n vertices by P n , C n and S n , respectively.
There are two categories of bases of bicyclic graphs, as described here. Denoted by ∞(p, l, q) the graph obtained from two vertex-disjoint cycles C p and C q by connecting one vertex u * of C p and one vertex v * of C q with a path P l+1 = u * · · · v * of length l (if l = 0, identifying u * with v * ), as depicted in Figure 1. Denoted by θ(a, b, c) the union of three internally disjoint paths P a+1 , P b+1 , P c+1 of length a, b, c (a, b, c ≥ 1 and at most one of them is 1) respectively with common end vertices u * and v * , as depicted in Figure 1. Notice that any bicyclic graph is obtained from a θ(a, b, c) or an ∞(p, l, q) by attaching trees to some of its vertices. The bicyclic graphs containing ∞(p, l, q) and θ(r, s, t) as its base are called ∞-graph and θ-graph, respectively.
A subset M ⊆ E(G) is called a matching of G if no pair of edges in M share a common vertex. The matching number a graph G is the maximum cardinality of a matching in G. If vertex x ∈ V(G) is incident with some edges of M, where M is a matching of G, then x is said to be M-saturated. M is called a perfect matching if each vertex of G is M-saturated. We can refer [20] for other terminologies and notations.

Preliminaries
Let S (x, y) = x y + y x , where x, y ≥ 1. One can easily get the Lemmas 2.1 and 2.2.
, where r, t ≥ 2 and r < t. Then l(t, r) is increasing for t and r, respectively.
Proof. It is evident that l(t, r) is increasing for r. Furthermore, since then l(t, r) is increasing for t.
Lemma 2.5. [14] Among the set of n-vertex (n ≥ 3) unicyclic graphs, the cycle C n is the unique graph with the minimum S DD index.

S DD S DD S DD index of unicyclic graphs with given matching number
For integers m ≥ 2, denoted by U U U n,m the set of n-vertex unicyclic graphs with matching number m. Let U U U * n,m be the unicyclic graphs on n vertices arisen from C 3 by attaching n − 2m + 1 pendant edges and m − 2 paths of length 2 to its one vertex, as depicted in Figure 2 Lemma 3.1. [21] Let U ∈ U U U 2m,m and T be a tree in U attached to a root r, where m ≥ 3. If y ∈ V(T ) is a vertex furthest from the root r with d U (y, r) ≥ 2, then y is a pendant vertex and adjacent to a vertex x of degree 2.
Thus for m = 2, the theorem is true. We assume that m ≥ 3 and the result holds for all unicyclic graphs on fewer than 2m vertices with a perfect matching. Suppose M is a perfect matching of U. If U C 2m , by Lemma 2.5, it follows that S DD(C 2m ) < S DD(U U U * 2m,m ) = f (2m, m). So we assume that U C 2m in the following proof. This implies that PV(U) ∅.
Let y ∈ PV(U), then U contains a tree T r attached on a root r ∈ V(C) such that y ∈ V(T r ), where C is the cycle of U. Let d T r (r, z) = max{d T r (r, y)|y ∈ V(T r )} and T T T U be the set of all pendant trees in U. We discuss in three cases.
In view of Lemma 3.2, then U is obtained from a cycle, say C s = x 1 x 2 · · · x s x 1 , by adding a pendant edge to some vertices on C s . If just one pendant edge is attached to every vertex of C s , then S DD Let y 2 be the pendant vertex adjacent to x 2 . Since U ∈ U U U 2m,m , it can be seen that a two-degree vertex can not be adjacent to two three-degree vertices.
By the definition of S DD index, induction hypothesis and Lemma 2.1, it follows that Assume without loss of generality that v 1 ∈ PV(U), By the definition of S DD index, induction hypothesis and Lemmas 2.1, 2.2, we have The equalities above hold only if S DD Then By the definition of S DD index, induction hypothesis and Lemma 2.1, it follows that Assume without loss of generality that By the definition of S DD index, induction hypothesis and Lemma 2.1, we have with equality if and only if U U U U * n,m . Proof. By induction on n. If n = 2m, by Theorem 3.4, the result holds. Now suppose that n > 2m. If U C n , it can be seen that n = 2m + 1. By Lemma 2.5, it follows that S DD(C 2m+1 ) < S DD(U U U * 2m+1,m ). The theorem holds. Thus we suppose that U C n in the following proof. By Lemma 3.3, it follows that there is a pendant vertex y and an m-matching M such that y is not M-saturated. Let xy ∈ E(U) and d U (x) = t. Let N U (x) ∩ PV(U) = {y 1 , y 2 , · · · , y r−1 , y r = y} and N U (x) \ PV(U) = {u 1 , u 2 , · · · , u t−r }. Then d U (u i ) ≥ 2 for each i = 1, 2, · · · , t − r. Furthermore, since U is a unicyclic graph and there exist at least m−2 M-saturated vertices in V(U)\{x, y 1 , y 2 , · · · , y r−1 , y r , u 1 , u 2 , · · · , u t−r }, then n = |V(U)| ≥ t+1+m−2, that is t ≤ n − m + 1. Let U = U − y. Then U ∈ U U U n−1,m . We discuss in two cases. Case 1. r = 1. Now, y = y 1 . By the induction hypothesis and Lemma 2.1, for n > 2m, it follows that Notice that there exist at least r − 1 vertices which are not M-saturated, then n − (r − 1) ≥ 2m, that is r ≤ n − 2m + 1. By the induction hypothesis and Lemmas 2.1, 2.4, it follows that With the equalities hold only if S DD(U ) = f (n − 1, m), t = n − m + 1, r = n − 2m + 1 and d U (u j ) = 2 for j = 1, 2, · · · , t − r, which implies that U U U U * n−1,m , and U U U U * n,m .
We assume that m ≥ 4 and the result holds for all bicyclic graphs on fewer than 2m vertices with a perfect matching. Suppose M is a perfect matching of B. For y ∈ PV(B), there exists a tree T r attached on a root r ∈ V (θ(a, b, c)) or r ∈ V(∞(p, l, q) in B such that y ∈ V(T r ), where T r is a pendant tree in B. Let d T r (r, z) = max{d T r (r, y)|y ∈ V(T r )} and T T T B be the set of all pendant trees in B. We discuss in three cases.