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The number of spanning trees in graphs or in networks is an important issue. The evaluation of this number not only is interesting from a mathematical (computational) perspective but also is an important measure of reliability of a network or designing electrical circuits. In this paper, a simple formula for the number of spanning trees of the Cartesian product of two regular graphs is investigated. Using this formula, the number of spanning trees of the four well-known regular networks can be simply taken into evaluation.

In this paper, we deal with simple undirected graphs having no self-loop or multiple edges and consider the Cartesian product of two regular graphs only. It is well known that, for designing large-scale interconnection networks, the Cartesian product is an important method to obtain large networks from smaller ones, with a number of parameters that can be easily calculated from the corresponding parameters for those small initial graphs. The Cartesian product preserves many nice properties such as regularity, transitivity, super edge-connectivity, and super point-connectivity of the initial regular graphs [

Alternatively, the study of the number of spanning trees in a graph has a long history and has been very active because the problem has different practical applications in different fields. For example, the number characterizes the reliability of a network and, in physics, designing electrical circuits, analyzing energy of masers, and investigating the possible particle transitions [

The number of spanning trees of some special network has been taken into evaluation [

The number of spanning trees of Boolean

Let

If

If

Let

(a) The coefficient of

On the other hand,

So we only need to prove that the coefficient of

By (

(b) So the coefficient of

Hence the theorem is proved due to (a) and (b).

Since a real symmetric matrix is with the property that the sum of its rows (and its columns) is zero, the rank of

If the eigenvalues of the Kirchhoff matrix

By Lemmas

Let the eigenvalues of the adjacency matrix

We know

We obtain

Hence the lemma is proved by Lemma

Let

Let

If

The lemma is easily obtained.

If the products

Let

If

Consider

If the points of

Since

where

where

Clearly

Let

By Lemmas

If

Since

Let

We know

If

By Lemma

The

When

We denote

The eigenvalues of

Since

If the distinct eigenvalues of the Kirchhoff matrix

where

Since

Hence if we take the eigenvalue 0 of

The main theorem in [

The number of spanning trees of

Since the degree of

The number of spanning trees of the Boolean

Since

The

Thus

The number of spanning trees of

Since the eigenvalues of

The number of spanning trees of

The generalized Boolean

Setting

The eigenvalues of the adjacent matrix

If

The eigenvalues of the adjacent matrix

Since

The main theorem in [

The number of spanning trees of

It follows that the points of

When

The hypercube network

The number of spanning trees of

It follows that the points of

Due to the high dependence of the network design and reliability problem, electrical circuits designing issue are on the graph theory. For example, the larger degree of points a network has, the more I/O ports and edges are needed and the more cost is required. The evaluation of this number not only is interesting from a mathematical (computational) perspective but also is an important issue on practical applications. However, the study for spanning trees of the Cartesian product of regular graphs remains an open and important invariant. In this paper, the eigenvalues of the Kirchhoff matrix of Cartesian product of two regular graphs,

The authors declare that there is no conflict of interests regarding the publication of this paper.

The authors would like to thank the anonymous reviewers for their valuable suggestions.