Upper and Lower Bounds for the Kirchhoff Index of the n-Dimensional Hypercube Network

Network is usually modelled by a connected graph G � (VG, EG) with order n, labeled as VG � v1, v2, . . . , vn 􏼈 􏼉 and EG � e1, e2, . . . , em 􏼈 􏼉. )e adjacency matrix A(G) of G is a square matrix with n vertices, in which elements aij are 1 or 0, depending on whether there is an edge or not between vertices i and j. )e degree diagonal matrix of G is denoted by D(G) � diag d1, d2, . . . , dn 􏼈 􏼉, where d1, d2, . . . , dn are the degree of vertices v1, v2, . . . , vn, respectively. Together with the adjacency and degree matrix, one arrives at the Laplacian matrix, whose expression can be written as L(G) � D(G) − A(G). For other notations and graph theoretical terminologies that not state here, we follow [1]. Various parameters are always used to characterize and describe the complex networks of which the fundamental one is named as the distance dij, concerned as the shortest path between the vertices i and j in networks. Similarly considering the distance dij, Klein and Randić in 1993 presented a novel distance function, named as resistance distance [2]. Denote rij the resistance distance between two arbitrary vertices i and j in electrical networks by replacing every edge by a unit resistor [3–7]. )e Kirchhoff index Kf(G) of networks is defined as


Introduction
Network is usually modelled by a connected graph G � (V G , E G ) with order n, labeled as V G � v 1 , v 2 , . . . , v n and E G � e 1 , e 2 , . . . , e m . e adjacency matrix A(G) of G is a square matrix with n vertices, in which elements a ij are 1 or 0, depending on whether there is an edge or not between vertices i and j. e degree diagonal matrix of G is denoted by D(G) � diag d 1 , d 2 , . . . , d n , where d 1 , d 2 , . . . , d n are the degree of vertices v 1 , v 2 , . . . , v n , respectively. Together with the adjacency and degree matrix, one arrives at the Laplacian matrix, whose expression can be written as L(G) � D(G) − A(G). For other notations and graph theoretical terminologies that not state here, we follow [1].
Various parameters are always used to characterize and describe the complex networks of which the fundamental one is named as the distance d ij , concerned as the shortest path between the vertices i and j in networks. Similarly considering the distance d ij , Klein and Randić in 1993 presented a novel distance function, named as resistance distance [2]. Denote r ij the resistance distance between two arbitrary vertices i and j in electrical networks by replacing every edge by a unit resistor [3][4][5][6][7]. e Kirchhoff index Kf(G) of networks is defined as e Kirchhoff index has attracted more and more attentions due to its practical applications in the fields of physical interpretations, electric circuit, and so on [8][9][10][11]. e Kirchhoff index of some product graphs, join graphs, and corona graphs were studied [5,7]. e more results of the applications on the Kirchhoff index were explored in [12][13][14].
In what follows, the rest of the context is summarized. Section 2 proposes the main definition and preliminaries in our discussion. Some bounds on the Kirchhoff index of hypercubes Q n are deduced in Section 3. We conclude the paper in Section 4.

Definition and Preliminaries
In this section, we recall some basic definition in graph theory. e hypercube network Q n may be constructed from the family of subsets of a set with a binary string of length n, by making a vertex for each possible subset and joining two vertices by an edge whenever the corresponding subsets differ in a single binary string. e hypercube network Q n admits several definitions of which one is stated as below [15]. e hypercube network Q n is repeatedly constructed by making two copies of Q n− 1 , written as Q 0 n− 1 and Q 1 n− 1 , respectively. Meanwhile, adding repeatedly 2 n− 1 edges as below, let e hypercube network Q n obtained more and more admirable concentrations due to its surprising properties, for instance, symmetry, regular structure, strong connectivity, small diameter, and so on [16,17]. For more results on the hypercube network and its applications, see [18][19][20][21].
Next, we recall the formula for the Kirchhoff index in the hypercube Q n with n ≥ 2.
Theorem 1 (see [3]). For the hypercube network Q n with n ≥ 2, where 2i(i � 1, . . . , n) is the eigenvalue of the Laplacian matrix of the hypercube network and the binomial coefficients n i are the multiplicities of the eigenvalues 2i.
e authors of [23] obtained a closed-form formula for the Kirchhoff index of the d-dimensional hypercube and found the asymptotic value 2 2 d /d by using probabilistic tools. e result of eorem 3 is obtained by directly calculating the eigenvalues of the Laplacian matrix of the hypercube network, which is different from the technique in [23].

Main Results
In this section, one will estimate the Kirchhoff index of n-dimensional hypercube, i.e., our goal is to estimate the quantity: .
Theorem 3. For the hypercube network Q n with n ≥ 2, then Consider that n i�1 n i By virtue of By means of calculating the right of equation (6), one can establish the following identity: Hence, Simply, from the left of the above inequality, we obtain Apparently, the left of the above inequality converges to the asymptotic value 2 2 d /d for large enough n. e proof of lower bound is completed.
For the upper bound, we have similar theorem to consider as follows.

Complexity
Kf Q n ≤ 2 n 2 n+1 − n − 2 n + 1 � 4 n n 2n n + 1 − n + 2 2 n (n + 1) . (15) e above estimate looks a little complicated. e upper bound is roughly twice the asymptotic value. Hence, a new upper bound is explored as follows.
Theorem 5. For the hypercube network Q n with n ≥ 2, Following the identity which is obtained in [24], Fixing x � 2, one arrives at n i�1 n i Namely, According to equation (19) and eorem 2, one obtains Using equation (20), one has On the contrary, Using eorem 2 and substituting equations (22) to (21), one obtains the desired result: is has completed the proof.

Further Discussion
We, at this place, try another way to estimate the Kirchhoff index of n-dimensional hypercubes.
Theorem 6. For the hypercube networks Q n with n ≥ 2, then Let S n � n i�1 C i n /i, then Consequently, One can easily check that S 1 � 1. Hence, S n − S n− 1 � (2 n /n) − (1/n).
By virtue of the above equality, we obtain e proof of eorem 6 is completed.

Data Availability
e data used to support the findings of this study are available within paper.

Conflicts of Interest
e authors declare no conflicts of interest.

Authors' Contributions
Data curation was carried out by J-B.L.; J-B.L. and J.Cao helped with the methodology; J.Z., Z-Y.S., and F.E. Alsaadi wrote the original draft. All authors read and approved the final manuscript.