The Kirchhoff Index of Folded Hypercubes and Some Variant Networks

The n-dimensional folded hypercube FQ n is an important and attractive variant of the n-dimensional hypercube Q n , which is obtained from Q n by adding an edge between any pair of vertices complementary edges. FQ n is superior to Q n in many measurements, such as the diameter of FQ n which is ⌈n/2⌉, about a half of the diameter in terms of Q n . The Kirchhoff index Kf(G) is the sum of resistance distances between all pairs of vertices in G. In this paper, we established the relationships between the folded hypercubes networks FQ n and its three variant networks l(FQ n ), s(FQ n ), and t(FQ n ) on their Kirchhoff index, by deducing the characteristic polynomial of the Laplacian matrix in spectral graph theory. Moreover, the explicit formulae for the Kirchhoff indexes of FQ n , l(FQ n ), s(FQ n ), and t(FQ n ) were proposed, respectively.


Introduction
It is well known that interconnection networks play an important role in parallel communication systems.An interconnection network is usually modelled by connected graphs  = (, ), where  denotes the set of processors and  denotes the set of communication links between processors in networks.In this work, we are concerned with finite undirected connected simple graphs (networks).For the graph theoretical definitions and notations, we follow [1].
Let  be a graph with vertices labelled 1, 2, . . ., .The resistance distances between vertices  and , denoted by   , are defined to be the effective electrical resistance between them if each edge of  is replaced by a unit resistor [2].A famous distance-based topological index as the Kirchhoff index Kf(), is defined as the sum of resistance distances between all pairs of vertices in .Define known as the Kirchhoff index of  [2].The Kirchhoff index attracted extensive attention due to its wide applications in physics, chemistry, graph theory, and so forth [3][4][5][6].For example, Zhu et al. [7] and Gutman and Mohar [8] proved that relations between Kirchhoff index of a graph and Laplacian eigenvalues of the graph.The Kirchhoff index also is a structure descriptor [9].
However, it is difficult to design some algorithms [7,10,11] to calculate resistance distances and the Kirchhoff indexes of graphs.Hence, it makes sense to find explicit closed form for some special classes of graphs, for instance, the Kirchhoff index for cycles and complete graphs [12] which has been computed, geodetic graphs [13], some composite graphs [14], and composite networks [15].Besides, many efforts were also made to obtain the Kirchhoff index bounds for some graphs [11,16] and characterize extremal graphs as well, such as bicyclic graphs and cacti graphs [6,17].Details on its theory can be found in recent papers [11,16,17] and the references cited therein.
The hypercube   is one of the most popular and efficient interconnection networks due to its excellent performance for some practical applications.There is a large amount of literature on the properties of hypercubes networks [18][19][20].
As an important variant of   , the folded hypercube networks   , proposed by El-Amawy and Latifi [18], are the graph obtained from   by adding an edge between any pair of vertices complementary addresses.The folded hypercube   , in which diameter of   is ⌈/2⌉, about half the diameter of   , has the same number of vertices as a hypercube   and 2 −1 edges more than hypercube; at the same time, the folded hypercubes preserve the symmetric properties of the hypercubes.The folded hypercubes   obtained considerable attention due to its perfect properties, such as symmetry, regular structure, strong connectivity, and small diameter, and many of its properties have been explored [21][22][23][24][25][26].
But few works appear on the Kirchhoff index for the combinatorial networks, such as hypercubes   and folded hypercubes   , except that Palacios and Renom [11] studied the bounds of the Kirchhoff index of hypercubes   by using probabilistic tools in 2010.In present paper, we established the relationships between the folded hypercubes networks   and three variant networks (  ), (  ), and (  ) on their Kirchhoff index, by deducing the characteristic polynomial of the Laplacian matrix in spectral graph theory.
Recall the definitions of n-dimensional folded hypercubes networks   as follows [18].Definition 1 (see [18]).The folded hypercubes   can be constructed from   by adding an edge to every pair of vertices with complementary addresses.Two vertices  =  1  2 ⋅ ⋅ ⋅   and  =  1  2 ⋅ ⋅ ⋅   are adjacent in the folded hypercubes   .
From above definition of   , it is easy to get that the folded hypercubes   have 2  vertices and (+1)2 −1 edges, respectively.
The line graph of a graph , denoted by (), is the graph whose vertices correspond to the edges of  with two vertices of () being adjacent if and only if the corresponding edges in  share a common vertex.The subdivision graph of a graph , denoted by (), is the graph obtained by replacing every edge in  with a copy of  2 (path of length two).The total graph of a graph , denoted by (), is the graph whose vertices correspond to the union of the set of vertices and edges of , with two vertices of () being adjacent if and only if the corresponding elements are adjacent or incident in .
Gao et al. [27] obtained special formulae for the Kirchhoff index of (), (), and (), where  is a regular graph.Motivated by above results, we present the corresponding calculated formulae for the Kirchhoff index of the hypercubes networks   and its three-variant networks (  ), (  ), and (  ) in this paper.
The remainder of present paper is organized as follows.Section 2 gives some basic notations and some preliminaries in our discussion.The proofs of our main results are in Section 3 and some conclusions are given in Section 4, respectively.

Notations and Some Preliminaries
In this section, we introduced some basic properties which we need to use in the proofs of our main results.
M. Chen and B. X. Chen have studied the Laplacian spectra of folded hypercubes networks in 2011 [21].
Lemma 2 (see [21]).For the folded hypercubes networks   with  ≥ 2, the spectrum of Laplacian matrix in terms of hypercubes networks is as follows.
(1) If  ≡ 0(mod2), (2) where    are the binomial coefficients; the elements in the first and second rows are the eigenvalues of the Laplacian matrix of folded hypercubes networks and the multiplicities of the corresponding eigenvalues.Lemma 3 (see [7,8]).Let  be a connected graph with  ≥ 2 vertices; then, Let  () () be the characteristic polynomial of the Laplacian matrix of a graph ; the following results were shown in [27].
It is worthwhile to note that the conclusion of Lemma 4 is not completely correct; the authors [28] recently show the Laplacian characteristic polynomial of (), where  is a regular graph, which corrects Lemma 3 in Gao et al. [27] as follows.
The following lemma gives an expression on Kf(()) and Kf() of a regular graph .
Lemma 6 (see [28]).Let  be a r-regular connected graph with  vertices and  edges and  ≥ 2; then, For proving the formula for the Kirchhoff index on the subdivision graph of hypercubes (  ), we prove the following lemma utilizing Vieta's theorem; in our proof, some techniques in [27] are referred.Lemma 7. Let  (  ) () be the characteristic polynomial of the Laplacian matrix of the folded hypercubes   with  ≥ 2 and where  2  −1 ,  2  −2 are the coefficient of  and  2 in the characteristic polynomial, respectively.

Main Results
3.1.The Kirchhoff Index in Folded Hypercubes Networks   .
In this section, we firstly give formula for the Kirchhoff index of the folded hypercubes   with any positive integer .

The Kirchhoff Index in the Line Graph of Folded
Hypercubes Networks (  ).In the following theorem, we proposed the formula for calculating the Kirchhoff index, denoted by Kf((  )), on the line graph of folded hypercubes (  ).
Proof.Now for convenience, we denote the numbers of vertices and edges in the folded hypercubes networks   by  = 2  and  = ( + 1)2 −1 , respectively.By Lemma 4, Notice that folded hypercubes networks   are regular graphs with the degree of any vertex being  =  + 1; compare the spectrum of   as follows.

The Kirchhoff Index in the Subdivision Graph of Folded
Hypercubes Networks (  ).In an almost identical way as Theorem 9, we derived the formula for the Kirchhoff index on the subdivision graph of folded hypercubes (  ), denoted by Kf((  )).

The Kirchhoff Index in the Total Graph of Folded Hypercubes Networks 𝑡(𝐹𝑄 𝑛
).We now proved the formula for the Kirchhoff index in the total graph of the folded hypercubes (  ), denoted by Kf((  )).
Theorem 11.Let (  ) be the total graphs of the folded hypercubes networks   with any positive integer ; then, (42)