New upper and lower bounds for the additive degree-Kirchhoff index

Given a simple connected graph on $N$ vertices with size $|E|$ and degree sequence $d_{1}\leq d_{2}\leq ...\leq d_{N}$, the aim of this paper is to exhibit new upper and lower bounds for the additive degree-Kirchhoff index in closed forms, not containing effective resistances but a few invariants $(N,|E|$ and the degrees $d_{i}$) and applicable in general contexts. In our arguments we follow a dual approach: along with a traditional toolbox of inequalities we also use a relatively newer method in Mathematical Chemistry, based on the majorization and Schur-convex functions. Some theoretical and numerical examples are provided, comparing the bounds obtained here and those previously known in the literature.


Introduction
The Kirchhoff index R(G) of a connected undirected graph G = (V, E) with vertex set {1, 2, . . . , N} and edge set E was defined by Klein and Randić [1] as where R ij is the effective resistance of the edge ij. This index has undergone intense scrutiny in recent years and researchers have come up with several modifications to it that take into account the degrees of the graph under consideration. On the one hand, Chen and Zhang defined in [2] the degree-Kirchhoff index as where d i is the degree (i.e., the number of neighbors) of the vertex i. This index has been studied in [3][4][5][6]. On the other hand, Gutman et al. defined in [7] the degree resistance index as and worked on the identification of graphs with lowest such degree among unicyclic graphs.
Perhaps it would be convenient to unify the nomenclature and call R * (G) the multiplicative degree-Kirchhoff index while calling R + (G) the additive degree-Kirchhoff index.
One fruitful viewpoint to this area is the probabilistic approach, that we have used in [3,[8][9][10][11], and can be summarized as follows: on the graph G we can define the simple random walk as the N -state Markov chain X n , n ≥ 0, with transition probability matrix P = (p ij ), 1 ≤ i, j ≤ N , whose entries are zero unless i and j are neighbors, in which case it is given by The stationary distribution π of this Markov chain, i.e., the unique left probabilistic eigenvector of P associated to the eigenvalue 1, is given by where |E| is the number of edges of G.
The hitting time T b of the vertex b is the number of jumps that the walk takes until it lands on b, and its expected value when the walk starts at a is denoted by E a T b . It is well known (see [8])that The purpose of this article is to give bounds for the additive degree-Kirchhoff index.
On the one hand, we give a simple upper bound that is shown to be attained, except for the constant of the largest term, by the symmetric barbell graph. On the other hand, we use probabilistic arguments in order to get a general formula for R + (G) in terms of Kemeny's constant and a sum of hitting times normalized by the stationary distribution, that have lower bounds known in the Markov chain literature; this way we obtain a lower bound for R + (G) that is attained by the complete graph.

The bounds
For the upper bound, it is immediate that using the upper bound for R(G), which occurs in the linear graph, obtained in [8]. The linear graph, however, with an additive degree-Kirchhoff index of order N 3 only, is a poor with the bN part and adding to (5), and given that all other contributions generate lower powers of N , we find that the coefficient of N 4 is given by Partial differentiation of the bivariate function F (a, b) given by (6) shows that its critical points are (0, 0) (corresponding to the linear graph, without N 4 term), (0, 1), (1, 0) (the complete graph, again, without N 4 term), (0, 1 2 ), ( 1 2 , 0) (the lollipop graph, for which R + (G) ∼ 1 16 N 4 ) and ( 1 3 , 1 3 ), (the symmetric ( 1 3 , 1 3 , 1 3 ) barbell graph, where the maximum is attained, with value R + (G) ∼ 2 27 N 4 ). We conjecture that this maximal value among all barbell graphs is indeed the maximum over all graphs on N vertices. Now we work on the lower bound. From (2), (3) and (4), it is clear that -653-It is well known (see [3,12]) that the sum j π j E i T j is a constant K independent of i, usually called Kemeny's constant, that can be expressed in terms of the eigenvalues λ i = λ 1 = 1 of the matrix P as It is also known (see [3]) that the multiplicative degree-Kirchhoff index can be written as Therefore showing a relationship between the two degree-Kirchhoff indices.
The sum i π i E i T j in general depends on j, but there is a well known lower bound (see [12]) stating that Inserting this into (7) we obtain Two applications of the harmonic mean-arithmetic mean inequality to equation (8), and the facts that N i=2 λ i = −1 and N j=1 d j = 2|E|, allow us to conclude that a bound that is attained by the complete graph K N , as an easy computation shows.