On the Inverse Degree Polynomial

: Using the symmetry property of the inverse degree index, in this paper, we obtain several mathematical relations of the inverse degree polynomial, and we show that some properties of graphs, such as the cardinality of the set of vertices and edges, or the cyclomatic number, can be deduced from their inverse degree polynomials.


Introduction
The interest in topological indices lies in the fact that they synthesize some of the fundamental properties of a molecule into a single value, and therefore, find many applications in chemistry. With this in mind, several topological indices have been studied so far; we note the seminal work by Wiener [1] in which he used the distances of a chemical graph in order to model the properties of alkanes. In particular, the inverse degree index ID(G) of a graph G is defined by where d u = deg G u denotes the degree of the vertex u in G.
The first time that the inverse degree of a graph attracted attention was due to numerous conjectures that were generated by the computer program Graffiti [2]. Since then, in many works by several authors (e.g., [3][4][5][6][7]), the relationship between other graph invariants, such as diameter, edge-connectivity, matching number and Wiener index, have been studied.
The study of polynomials in graphs is a very useful subject from a theoretical and practical point of view and is of growing interest. In particular, in [8], there is an extensive development of the domination polynomial, while in [9], we find a study of the roots of independent polynomials. In both cases, the fundamental idea was to have a characterization of the graphs from the polynomials.
In the work by Baig et al. [10], the authors studies Omega, Sadhana, and PI counting polynomials and computed topological indices associated with them. These polynomials are used to predict several physico-chemical properties of certain chemical compounds. The same three polynomials were computed in a more recent work by Imran et al. [11]. Here the authors computed polynomials for mesh-derived networks.
We want to recall that polynomials in graphs have been widely used to study the structural properties of certain graph families. Several graph parameters have been used to define a graph polynomial, for instance, differential number, the parameters associated to matching, independent and domination sets, chromatic numbers and many others (see, e.g., [12] and the references therein). In recent years there have been many works on graph polynomials associated with different topological indices (see, e.g., [13][14][15][16][17]).
In [13], Shuxian defined the Zagreb polynomial of a graph G as In [14] the harmonic polynomial of a graph G is defined as The harmonic polynomials of the line graphs were studied in [17]. The inverse degree polynomial is studied to understand better the inverse degree topological index. In [15], this polynomial was used in order to obtain bounds of the harmonic index of the main products of graphs; in order to do that, a main tool was the inverse degree polynomial (or ID polynomial) of a graph G, defined as Thus, we have 1 0 ID(G, x) dx = ID(G). So, one can expect to obtain information on the inverse degree index from the properties of the inverse degree polynomial. Note that, x(xID(G, x)) = M * 1 (G, x).

Some Preliminaries
The next interesting result appears in [15].

Proposition 1.
If G is a graph with n vertices, and k of them are pendant vertices, then: if and only if G is not isomorphic to an union of path graphs P 2 , • ID(G, x) is strictly convex on [0, ∞) if and only if G is not isomorphic to an union of path graphs and/or cycle graphs, and The following results are direct.

Proposition 2.
If G is a k-regular graph with n vertices, then ID(G, x) = nx k−1 .
The following result computes the ID polynomial of: K n (the complete graph with n vertices), C n (the cycle with n 3 vertices), Q n (the n-dimensional hypercube), K n 1 ,n 2 (the complete bipartite graph with n 1 + n 2 vertices), S n (the star graph with n vertices), P n (the path graph with n vertices), W n (the wheel graph with n ≥ 4 vertices), and S n 1 ,n 2 (the double star graph with n 1 + n 2 + 2 vertices).
The forgotten topological index (or F-index) is defined as

Main Results
The next result allow us to obtain information about the graph using the ID polynomial.

Proposition 4.
If G is a graph with n vertices and m edges, then we have Proof. The equality ID(G, 1) = n is a consequence of Proposition 1. By the handshaking Lemma, we have and

Corollary 1.
If G is a graph with n vertices and m edges, then we have Proposition 2 shows that any two k-regular graphs with the same cardinality of vertices, have the same ID polynomial. The following question is very natural: How many graphs can be characterized by their ID polynomials? Although this is a very difficult question, Corollary 1 provides a partial answer: Graphs with different cardinality of vertices or edges have different ID polynomials. This property allows to conclude the following.

Corollary 2.
If Γ is a proper subgraph of the graph G, then ID(Γ, x) = ID(G, x).
Also, we can obtain information about the cycles in the graph by using the ID polynomial. The cyclomatic number of a connected graph with n vertices and m edges is defined as γ(G) = m − n + 1.

Proposition 5.
If G is a connected graph, then its cyclomatic number is In particular, if ID(G, 1) = ID (G, 1) + 2, then G is a tree.
Furthermore, Proposition 6 and Theorem 7 show that two graphs with the same ID polynomial have to be similar, in some sense.
If k is any positive integer, we consider Vieta's formulas give Note that a k,0 = (−1) k k! and a k,k−1 = − 1 2 k(k + 1). The following result generalizes Proposition 4. Proposition 6. If G is a graph and k is a positive integer, then The next result gives bounds of the ID index in terms of the values of the ID polynomial at the points 0, 1/2 and 1.

Proposition 7.
For any graph G, we have the inequalities and the equality in each inequality is attained if and only if G is isomorphic to an union of path graphs.
Proof. Hermite-Hadamard's inequality gives for every convex function g on [0, 1], and both inequalities are strict when g is strictly convex. If G is not isomorphic to an union of path graphs and/or cycle graphs, then Proposition 1 gives that ID(G, x) is a strictly convex function. So, applying Hermite-Hadamard's inequality to the function g(x) = ID(G, x), we obtain If G is isomorphic to an union of path graphs and/or cycle graphs, then the ID polynomial of G has degree at most 1, and so, both inequalities are attained.

Given a graph G and a vertex
Given a polynomial P(x), let us denote by Deg P(x) the degree of P(x), and by Deg min P(x) the minimum degree of the monomials with non-zero coefficients of P(x).

Proposition 8.
Let G be a graph with n vertices, maximum degree ∆ and minimum degree δ. Then:

•
Deg ID(G, x) = ∆ − 1. Proof. If δ > 1 and G is regular, then Proposition 2 gives that x = 0 is the unique zero of ID(G, x). Assume now that x = 0 is the unique zero of the ID polynomial of the graph G. Thus, ID(G, x) = ax b−1 for some positive integers a, b, and d u = b for every u ∈ V(G), and so, G is regular. Also, Proposition 8 gives δ > 1.
The next result allows to obtain information about the connectedness and diameter of a graph G in terms of the degree of its ID polynomial.
Hence, diam G 2 and, consequently, G is a connected graph.
The following classical result gives an asymptotically sharp upper bound for the diameter of a connected graph (see [18] [Theorem 1]). Recall that, if t is any real number, then t denotes the lower integer part of t, i.e., the largest integer less than or equal to t.

Theorem 3 (Erdös, Pach, Pollack, and Tuza). Let G be a connected graph with n vertices and minimum degree
The two next results in [19] provide better estimations for the diameter of a connected graph.

Theorem 4.
If G is a connected graph with n vertices and minimum degree δ, then diam G ≤ n − 1 if δ = 1, and for every δ ≥ 2.
Theorem 5. Let n be the number of vertices of the connected graph G and with a minimum degree equal to δ.
These two results in [19] have the following consequences for general graphs.

Proposition 9.
If G is a graph with n vertices, minimum degree δ and at least r 1 connected components, and Γ is a connected component of G, then: Proof. Each connected component of G has at least δ + 1 vertices. Since G is a graph with at least r 1 connected components, G has at least r(δ + 1) vertices, i.e., r(δ + 1) n. Also, any connected component of G has at most n − (r − 1)(δ + 1) vertices.
If δ ≥ 2, then Theorem 4 gives The arguments in the proof of Proposition 9 have the following consequence.
Corollary 3. Let G be a graph with n vertices and minimum degree δ. Then G has at most n δ+1 connected components. In particular, if δ > n 2 − 1, then G is connected.

Proposition 10.
Let G be a graph with n vertices, minimum degree δ and at least r 1 connected components, and let Γ be a connected component of G.
Proof. The arguments in the proof of Proposition 9 give that Γ has at most n − (r − 1)(δ + 1) vertices.
The results that we are going to show in the following three propositions provide information about a graph based on its polynomial ID. Proposition 11. If G is a graph with at least r 1 connected components, and Γ is a connected component of G. Proof. Using the results summarized in Proposition 1, we have that the number of vertices that have degree 1 is equal to ID(G, 0), while ID(G, 1) is equal to the number of total vertices of the graph. On the other hand, it is obtained directly that if ID(G, 0) is different from zero then the minimum degree δ of the graph G is exactly 1, while if ID(G, 0) = 0 then the δ ≥ 2. Finally, from Proposition 8 we have Deg ID(G, x) = ∆ − 1 and Deg min ID(G, x) = δ − 1, and directly applying the results of Proposition 9 we obtain the proposed result.
Continuing with the same arguments used previously, from Proposition 1 we have that ID(G, 1) = n, while from Proposition 8 we have that Deg min ID(G, x) + 1 = δ, then as direct consequence of Corollary 3 we obtain the following result.
Proposition 12. Let G be a graph. Then G has at most ID(G, 1) Deg min ID(G, x) + 2 connected components. In particular, if 2 Deg min ID(G, x) + 4 > ID(G, 1), then G is connected.
In the same way, the next proposition is a direct consequence of the results given in Propositions 1, 8, and 10. Proposition 13. Let G be a graph with at least r 1 connected components, and let Γ be a connected component of G.
If p(x) is a polynomial, then we denote by D(p(x)) the number of non-zero coefficients of p(x). In particular, if p(x) = ID(G, x), then D (ID(G, x)) is the cardinality of the set {d u : u ∈ V(G)}.
Let us define inductively the graph Γ k for k 2. Let Γ 2 and Γ 3 be the path graphs P 2 and P 3 , respectively. If k 4, then we define Γ k from Γ k−2 in the following way: Choose two points u 1 , u k−1 / ∈ V(Γ k−2 ) and define Γ k by Theorem 6. If G is a graph with n vertices, minimum degree δ and at least r connected components, then the following statements hold: Proof. If G is a connected graph, then the inequalities δ d u n − 1 for every u ∈ V(G) give 1 D(ID(G, x)) n − δ. Assume now that G is not connected and let Γ be a connected component of G. The argument in the proof of Proposition 9 gives that Γ has at most n − (r − 1)(δ + 1) vertices. Thus, δ d u |V(Γ)| − 1 for every u ∈ V(Γ), and so, δ d u n − 1 − (r − 1)(δ + 1) for every u ∈ V(Γ), and we conclude 1 D(ID(G, x)) n − δ − (r − 1)(δ + 1).

Corollary 4.
If G is a graph with n vertices, then 1 D (ID(G, x)) n − 1.
The argument in the proof of Theorem 6 also gives the following result.
Next, we present a result that allows to bound the inverse degree index of a graph by using information about its ID polynomial.
Given a graph G, let us denote by k ∆ and k δ the cardinality of the sets {u ∈ V(G) : d u = ∆} and {u ∈ V(G) : d u = ∆}, respectively.

Proposition 15.
If G is a graph with n vertices, maximum degree ∆ and minimum degree δ, then and the equality in each inequality is attained if and only if we have either that G is regular or k δ + k ∆ = n. Hence, If G is regular, then the lower and upper bounds are both equal to n/δ, and the equalities hold. If k δ + k ∆ = n, then the lower and upper bounds are both equal to k δ /δ + k ∆ /∆, and the equalities hold.
Assume now that the equality is attained in the lower bound and G is not a regular graph. Thus, and so, k j = 0 for every δ < j < ∆. Hence, k δ + k ∆ = n.
If the equality is attained in the upper bound and G is not a regular graph, a similar argument also gives k δ + k ∆ = n.
If G is a regular graph is regular, then the both bounds are the same, and they are equal to GA 1 (G). If we have the equality, then 4(d u + d v ) −2 = ∆ −2 for every uv ∈ E(G); thus, G is a regular graph.

Conclusions
In this work we obtain several mathematical relations of the inverse degree polynomial from the symmetry property of the inverse degree index. We show that some properties of graphs, such as the cardinality of the set of vertices and edges, or the cyclomatic number, can be deduced from their inverse degree polynomials. We also obtain bounds of the ID index in terms of the values of the ID polynomial and we have a relation between the harmonic and the inverse degree polynomials.