Characteristic polynomials, spectral-based Riemann-Zeta functions and entropy indices of n-dimensional hypercubes

Abstract We obtain the characteristic polynomials and a number of spectral-based indices such as the Riemann-Zeta functional indices and spectral entropies of n-dimensional hypercubes using recursive Hadamard transforms. The computed numerical results are constructed for up to 23-dimensional hypercubes. While the graph energies exhibit a J-curve as a function of the dimension of the n-cubes, the spectra-based entropies exhibit a linear dependence on the dimension. We have also provided structural interpretations for the coefficients of the characteristic polynomials of n-cubes and obtain expressions for the integer sequences formed by the spectral-based Riemann-Zeta functions. Graphical abstract

characteristic polynomials as well as the spectral-based Riemann-Zeta functions. Both spectral based indices and coefficients in the characteristic polynomials are shown to form interesting sequences.
2 Mathematical and computational methods for characteristic polynomials, spectral-based Riemann-Zeta functional indices and entropies of nD-hypercubes

Characteristic polynomials
An n-dimensional hypercube graph, Q n , can be recursively generated by the Cartesian product [43] where K 2 is a graph with two vertices joined by an edge. Consequently, the adjacency matrix of Q n , A Qn , is given by where I is simply an identity matrix of order 2 n−1 × 2 n−1 .
One can construct adjacency matrices of nD-hypercubes using the above recursive relation. It may be recalled that the number of q-hyperplanes of a n-cube where 0 ≤ q ≤ n − 1 is given by the diagonal elements of the Coxeter configuration matrix of the n-cube shown below: Hence the no of vertices (q = 0) and the number of edges of an n-cube (q = 1) are given by 2 n and n × 2 n−1 , respectively. For example, for a 23-cube, the number of vertices and edges are 8,388,068 and 94,468,992. Thus computing the characteristic polynomials and spectral-based indices can be computationally challenging.
In the current study, we make use of the author's previously developed computational techniques [44][45][46] that invoke recursive Hadamard transform, Frame's method for the characteristic polynomials and their powerful computations extensions that employ quadruple precision arithmetic and dynamic computer. The Frame's method simplifies the computationally intensive generation of the characteristic polynomials to simply taking traces of matrix products constructed iteratively. This, in effect, reduces the O(n!) problem to O(n 3 ) computations. We start with the adjacency matrix constructed recursively. Subsequently, as described in depth in previous studies [44,45] we construct iterative matrix products and in each iteration the traces of the matrix products are taken. These traces contain the coefficients of the characteristic polynomials. A primary computational difficulty that remains is that the order of the adjacency Q n = Q n−1 × K 2 for n ≥ 2 with Q 1 = K 2 , matrices of n-cube grows as 2 n × 2 n and thus even with quadruple precision arithmetic only the first several coefficients can be evaluated. Consequently, we use the spectra of the adjacency matrices independently computed using recursive Hadamard transforms to compute the last three coefficients. If {λ 1 , λ 2 ,…,λ m } is the set of eigenvalues of n-cubes where m = 2 n then it can be shown that the constant coefficient C o , and the coefficients of x 2 , C 2 , are given by the following expressions.
where in the expression for C 2 � i … � j means omission of these two eigenvalues in the product where j > i. The above expression is a S-function obtained by omitting two terms at a time, and thus the symmetric sum is over m 2 such terms. Likewise the C 4 coefficient is obtained from the S-function that omits four terms l such that l > k > j > i, and thus the C 4 coefficient contains m 4 terms in the S-function, or 2 n × (2 n − 1) × (2 n − 2) × (2 n − 3)/24 terms as a function of n, the dimension of the n-cube. In general, it is not computationally feasible to generate all of the coefficients of the characteristic polynomials of n-cubes using such a S-function approach due to combinatorial explosion in the number of terms in the S-function. However, the technique generates the last three coefficients without too heavy load of computations for the n-cubes. Consequently, we generate the first few coefficients using the Frame's method-based computer codes while the last few coefficients were computed using the S-function based computer codes developed here. The eigenvalues themselves were computed using the recursive Hadamard transform in conjunction with the Givens-Householder tridiagonalization algorithm. The related matching polynomials of graphs are of considerable interest [47][48][49], and in the results and discussion section we explore this connection.

Riemann-Zeta function based spectral indices
The Riemann-Zeta function is of prime importance in number theory as it is the generating function for the distribution of prime numbers. Although it was introduced primarily in number theoretic context, it is an important quantifier for the spectral eigenvalues as we can obtain the spectral-based Riemann-Zeta functional Indices. The function is originally defined as where x is a variable complex number.
One can introduce the corresponding Riemann-Zeta function for the graph spectra, as it has been done previously for the Laplacians of graphs [50] with the following definition: where if the variable x is positive, the function becomes undefined for the case λ i = 0, and thus we omit this case for n-cubes. However, it is never an issue for negative values of x or for any n-cube when n is odd, as odd-dimensional hypercubes do not exhibit zero eigenvalues. Thus for even n-cubes, the zero eigenvalues are omitted in the sum when x is positive. Likewise as we will see in the ensuing section spectra-based entropy definition also eliminates zero eigenvalues.
The Riemann-Zeta functions computed for negative integral values of x define the spectra-based Riemann-Zeta functional indices. For example for x = − 2, the second order Riemann-Zeta functional index is given by which is simply the sum of the squares of the spectra. Likewise one can obtain sums of powers of the spectra for various integers. These set of spectral-based Riemann-Zeta functional Indices can then be employed as descriptors of graphs and molecular structures. For n-cubes considered here, we have obtained the spectral-based Riemann-Zeta functional Indices for up to 20th power. We have computed these indices for up to 23-cubes. For positive integral values also the Riemann-Zeta functional Indices can be obtained; a special case for the Laplacians with x = − 1 we obtain the Kirchhoff index. Consequently, the spectral-based Riemann-Zeta functional Indices are quite general and contain combinatorially rich structural information including the number of spanning trees contained in the graph and so forth.

Spectral-based entropies
Spectral-based entropy function can be defined analogous to the general definition of Shannon for the entropy of any function which in turn is defined by a summation series on the graph. For this purpose one can take the well-known graph energy function as the defining function for the spectral-based entropies. The graph energy is defined once we have the spectra of n-cube as Consequently, the spectral-based entropies can be obtained omitting zero eigenvalues as where GE is the graph energy as define above and the sum in the entropy definition explicitly omits the zero eigenvalues. This could mean an odd-even alternation in the entropy, as n-cubes with odd n do not contain zero eigenvalues while even n hypercubes contain zero eigenvalues. However, in practical computations, as we demonstrate the omission of zero eigenvalues makes no difference whatsoever in the entropy trends. The actual computation of GE and section on the spectral-based entropies for n-cubes can be further simplified by explicit consideration of the binomial distribution exhibited by the spectra of n-cubes. We exploit this property in all of the computations thus facilitating the computations of GEs and spectral-based entropies for up to 23-cubes.

Characteristic polynomials and spectra of n-cubes
Before we discuss the characteristic polynomials, we briefly consider the spectra of the n-cubes. We have made use of recursive Hadamard transform as discussed earlier [22]. For n-cubes the recursive Hadamard transforms enable rapid computations of the spectra of up to the 23-cube. In Table 1 we have shown selectively the numerical results for all hypercubes from the 10-cube to 23-cube. In a previous study the spectra of all n-cubes were shown [22]. It can be seen that the computed spectra follow a binomial spectral distribution (1 + λ) n for the n-cubes, where the coefficient of λ k gives the number of times the eigenvalue − n + 2k appears in the spectra. The computed graph energies as defined in the previous section are also included in Table 1.
Although from the spectra one can infer the overall form of the characteristic polynomial of an n-cube by the following expression, the computations needed for all the coefficients can be enormous. These computations of the polynomial shown below by expanding the all the binomials and multiplying them together to gather the coefficient of the various terms is a computationally daunting task. However, the above expression can be used to collect the coefficients of the constant term, l 2 terms and the first two coefficients as well. Beyond that it is not feasible to employ the above expression to compute the coefficients of the characteristic polynomials.
The coefficients of the characteristic polynomials of the n-cube shown in Tables 2, 3, 4, 5 and 6 contain rich molecular structural information as can be analysed through Sach's theorem. According to Sach's theorem, the kth coefficient, C k , in the characteristic polynomial is given by the summation series, where G i is the set of all Sach's subgraphs of G(n-cube) containing k vertices, r(g) being the number of cycles in g and c(g) is the number of disjoint components in g. It is noted that the Sach subgraphs include all components that contain vertices such that the components are either dimers or closed circuits of various lengths; they encompass k vertices of the parent graph G. The collection of Sach's subgraphs includes both the null set and the parent graph of n-cubes. Hence the C 4 coefficient for instance would include all squares and all possible disjoint dimers of the n-cubes.
The coefficient of x n−4 enumerates Sach's subgraphs containing 4 vertices in the n-, and these are two disjoint dimers in the n-cube and all cycles of length 4. As cycles of length 4 in n-cubes are also the number of faces of n-cubes, it can be inferred from Coxeter's general expression as Consequently the number of 2-matchings or the number of ways to place two disjoint dimers (M 2 ) on the n-cube is given by Thus for example, the M 2 coefficients are enumerated as 400, 2840, 17376, 97440, 516608, 2634624, 13058560, 63320576, 301707264, 1417009152, 6575120384, and 30195425280, for 4-cube through 13-cube, respectively as deduced from Tables 2, 3, 4 and 5. The matching polynomials are of considerable interest over the years [22,[47][48][49] as they pose significant challenges for computing.
The coefficient of x n−6 in the characteristic polynomial of n-cube yields the number of three disjoint dimers, a squarE+dimer not sharing any vertex, and a cycle of containing six vertices. The cycles of length 6 are generated by taking two adjacent squares of the n-cube and removing the edge common to the two squares. Hence we have the formulae for C 6 as It is evident that C(4; 2) is enumerated by the product of section the number of squares and number of edges but omitting the edges of the squares themselves so that the square and the chosen edges are disjoint. This is given by (n)(n − 1)2 n−2 × {n2 n−1 − 4}.
The above process can be iterated to provide structural interpretations for the various coefficients of the characteristic polynomials shown in Tables 2, 3, 4 and 5. It is noted that the constant coefficient of the characteristic polynomial of a n-cube is the product of the eigenvalues. This term is considerably more complex for the n-cubes, as it not only contains full dimer covers for the n-cube but also contributions from numerous other components in Sach's subgraphs with 2 n vertices. The constant coefficients of the n-cube characteristic polynomial can be readily factored using the eigenvalues and their frequencies which follow a binomial distribution.
The coefficients of various terms of the characteristic polynomials of the n-cubes form interesting integer sequences. The constant terms of the n-cubes exhibit odd-even alternation with the coefficients of n-cubes wit even n being zeroes while the odd n-cubes are not zeroes. Thus we have as the sequence for the constant coefficients for cube, 4-cube, 5-cube,…,13-cube, respectively.
Likewise, the sixth coefficient, C n−6 of the n-cube form the sequence The other interesting sequence is that of the last but one coefficient of the characteristic polynomial which is alternating in nature; given by

Spectral-based Riemann-Zeta functional indices of n-cubes
In Table 6 we have shown our computed Riemann-Zeta functional indices of n-cubes with n = 4 to 23. The spectra-based Riemann-Zeta functional Indices provide additional measures for the characterization of structures besides the characteristic polynomials. For n-cubes as seen from Table 6, all spectral-based Riemann-Zeta functional indices are integers, given that the n-cubes exhibit integral spectra. A few trends can be readily seen. For example, the second-order Riemann-Zeta index for a n-cube is given by n × 2 n , a trend that is followed by all n-cubes as seen from Table 6. All the odd-order Riemann-Zeta functional indices of n-cubes are zero because an n-cube is a regular graph and all regular graphs satisfy this relation. The fourth-order Riemann-Zeta functional indices satisfy the relation: where RZ 2 is the second-order (k = 2) Riemann-Zeta index. Likewise Rz 6 = a n × RZ 2 , where a n forms the sequence: The above sequence fits into the following expression in terms of n:  with n varying from 3 to 23. Based on the above pattern the RZ 6 for any n-cube can be obtained as: Likewise each of the Riemann-Zeta functional indices shown in Table 6 for n-cubes form an interesting sequence. There is a clear odd-even alternation in the number sequence for RZ 6 . Interestingly the sequence derived from RZ 6 ends with 1 for all odd n-cubes, a result not known up to now.

Spectral-based entropies of n-cubes
The computed graph energies and spectral-based graph entropies of n-cubes are shown in Table 7. Graph energies have been the topic of considerable interest over the years [39,40] as they seem to be one of the important spectra-based quantifies of graphs. Kaatz and Bultheel [40] have shown how graph entropies and graph energies can be used together to compute graph-based free energies and other properties to predict the properties of nanoclusters and hypercube-based clusters. Consequently, we have computed the graph energies and entropies for hypercubes with dimensions 3 to 23. As seen from Table 7, graph energies of all n-cubes are integers, as n-cubes exhibit integral spectra. Figure 1a and b show the variations of graph energies and graph entropies of n-cubes, respectively. While the graph energies vary exponentially in the form of a J-curve shown in Fig. 1a, the graph entropy is a linear function of the dimension of the n-cube (see Fig. 1b). Although the zero eigenvalues were omitted in the computations of graph entropies of n-cubes, the overall trend does not exhibit any odd-even alternation. As seen from Fig. 1b, the graph entropies of n-cubes exhibit a uniformly linear trend. The only exception is seen in the form of a small kink in transitioning from n = 4 to 5, mainly due to the omission of zero eigenvalues which show up as a small kink for smaller values of n while the trend becomes a uniform straight line for larger values of n.

Conclusion
In retrospect, we have obtained the characteristic polynomials, spectral-based Riemann-Zeta functional indices, graph energies and entropies of n-cubes. We have demonstrated that rich structurally based combinatorial information is contained in the coefficients of n-cubes. It was also shown that the Riemann-Zeta functional indices form very interesting integer sequences, some of which were analysed and fit into analytical expressions. While the spectral-based entropies exhibit a linear dependence on the dimension of the hypercubes, the graph energies exhibit an exponential J-curve as a function of the dimension of the n-cubes. The computations of the characteristic polynomials of n-cubes together with their spectral-based indices could become integral part of recursive logic, machine learning and artificial intelligence techniques as they pertain to big data, neural networks and large biological networks such as the genetic and phylogenetic networks. It is hoped that the present study would stimulate such future studies and applications of recursive logic methods and hypercubes. We also note that the reaction and isomerization graphs of interest in dynamic stereochemistry and dynamic NMR spectroscopy exhibit interesting topological patterns and Borel field patterns [25], and thus there is considerable scope in applying the techniques developed here to molecular dynamics of fluxional molecules. We anticipate the techniques developed here to be extended to donut-shaped benzenoids [51], holey nanographenes [34] and other nanomaterials in the future. Graphic arts for this work was prepared from Tom Ruen, open source, copy-right free, Public Domain Image [52].