Abstract
The number of spanning trees in a network determines the totality of acyclic and connected components present within. This number is termed as complexity of the network. In this article, we address the closed formulae of the complexity of networks’ operations such as duplication (split, shadow, and vortex networks of ), sum (, , and ), product ( and ), semitotal networks ( and ), and edge subdivision of the wheel. All our findings in this article have been obtained by applying the methods from linear algebra, matrix theory, and Chebyshev polynomials. Our results shall also be summarized with the help of individual plots and relative comparison at the end of this article.
1. Introduction
Preliminarily, the finite, connected, and simple network shall be our consideration. The complexity (number of spanning trees in a network) is an enormously useful network invariant and is significantly studied in algebraic graph theory, combinatorics, and networking. It admits main connections to computer networking and certain branches of engineering that are related to urban planning (civil engineering) specifically. In fact, the more rigidity and accuracy of a network is ensured by a greater number of spanning trees. Thus, the complexity refers to more perfectness and quality in a network. For further details of its applications, see [1–6].
No general complexity function for an infinite family of networks has been obtained in the literature yet. However, there is a scope of determining closed formulae to find out the complexity of networks of order , especially when the value of is sufficiently large. With the increase in the order of a network , it is significant to recapitulate the trend of its complexity function . Among the foremost of such findings, the credit of determining the complexity function of , as goes to Cayley [7]. In the same article, another very prominent result proposed by him is the complexity of the complete bipartite network which he enumerated as . The expression for the complexity of Mobius ladder is derived to be , for [8].
Recently, determining the number of spanning trees of networks using the determinants of certain matrices has become a hot topic. The most famous among these is Kirchhoff’s matrix tree theorem [4]. The aforesaid famous theorem states that the complexity of a network is a cofactor of its Kirchhoff’s matrix, where the Kirchhoff’s matrix of a network is given as .
The contraction-deletion theorem is a combinatorial technique to enumerate the number of spanning trees of a network. Let , then the complexity of can be calculated iteratively by usingwhere is obtained by contracting in until the two vertices and coincide and is just the edge deletion with respect to the edge [9].
The Laplacian spectrum is also helpful for the calculation of the complexity of the network . Let be the eigenvalues of the Laplacian matrix of . In [10], it is shown that
This method is a complicated one, and complexity of larger networks becomes cumbersome to calculate in this method. Temperley has proven that , where is the matrix with all 1 entries. Some recent work on the enumeration of the complexity of various families of networks can be found in [2, 9, 11–15].
1.1. Definitions and Preliminaries
Temperley’s equation mentioned above straightforwardly gives us the following lemma.
Lemma 1. (see [16]). Let be an -order network, thenwhere represents the complement of the network .
The above formula attains significance while enumerating the complexity as it represents as the determinant of a specific matrix rather than eigenvalues or cofactor.
The first-kind Chebyshev polynomials are defined as the solution of the iterative relation.
The standard solution of the above iterative relation gives
The second-kind Chebyshev Polynomials are defined as the solution of the following iterative relation:
The standard solution of the above iterative relation gives
Identity (7) is valid for all complex values of except .
Both first- and second-kind Chebyshev polynomials have a strong link with the determinants [17].
Lemma 2. (see [18, 19]).(i), , where(ii), , , where(iii); , where(iv), , , where
Lemma 3. (see [20]). and , , where is an circulant matrix given as
Lemma 4. (see [21]). Let , , , and be the block matrices of orders , , , and , respectively. Then,where and are nonsingular matrices.
Throughout the article, will represent the matrix of order . and will represent the row and column in a matrix, respectively. Also, the set of determinant operations () performed in Theorem 1 shall be used frequently in all our results.
2. Main Results
The realm of generating new structures by applying general operations to the existing ones always remains open in networking. In this section, we shall present our main findings consisting of the enumerated closed formulae of the generalized operation on certain networks. The necessary definition of a network’s operation [22, 23] shall be provided before the respective result.
Theorem 1. For all , the complexity of the network is given by
Proof. Consider the network with and , see Figure 1.
Applying Lemma 1, we haveNow, we perform the following operations simultaneously on the above determinant:(i)Adding all columns to (ii)Taking n + 5 common from . (iii)Subtracting from all columns(iv)Expanding along These operations yieldBy using Lemma 4, we obtainUsing Lemma 3, we have
Definition 1. The shadow of a network is obtained by taking another copy of , say , and then by making all those vertices adjacent to the corresponding adjacent vertices .
Theorem 2. For all , the complexity of the network is given by
Proof. Consider the network with and , see Figure 2.
Applying Lemma 1, we havewhere and represent the degree and adjacency matrices of the network , respectively.By using Lemma 4, we obtain
Definition 2. The splitting network of a network is obtained by taking a vertex set, say , corresponding to and then by making all those vertices adjacent to the corresponding vertices that preserve the adjacency of .
Theorem 3. For all , the complexity of the splitting network is given by
Proof. Let us consider the network with and , see the general formation in Figure 3.
Applying Lemma 1, we havewhere and represent the degree and adjacency matrices of the splitting network , respectively.By using Lemma 4, we obtain
Definition 3. The strong product of two networks and is a new network with vertex and edge sets as follows:.
Theorem 4. For all , the complexity of the strong product is given by
Proof. Consider the network with and , see its general formation in Figure 4.
Applying Lemma 1, we havewhere and represent the degree and adjacency matrices of the network , respectively.By using Lemma 4, we obtainCalculating the values of for We obtain . Upon generalizing for , we obtain .Finally, we obtain
We define here a new operation namely vortex of a network .
Definition 4. The vortex of a network is obtained by taking a vertex set, say , corresponding to and then by making all those vertices adjacent to those vertices that do not preserve the adjacency of .
Theorem 5. For all , the complexity of the vortex is given by
Proof. Consider the network with and , see Figure 5.
Applying Lemma 1, we havewhere and represent the degree and adjacency matrices of the vortex , respectively.By using Lemma 4, we obtainApplying Lemma 3,
Theorem 6. For all , the complexity of the network is given by
Proof. Consider the network with and , see Figure 6.
Applying Lemma 1, we haveBy using Lemma 4, we obtain
Theorem 7. For , the complexity of an edge-subdivision of the wheel is given by
Proof. Consider the edge-subdivision of wheel with , , see Figure 7.
Applying Lemma 1, we havewhere and represent the degree and adjacency matrices of the network , respectively.By using Lemma 4, we obtainUsing Lemma 2,
Theorem 8. For , the complexity of the rooted product. . is given as
Proof. Consider the network with and . The general formation is shown in Figure 8.
Applying Lemma 1, we havewhere and represent the degree and adjacency matrices of the network , respectively.By using Lemma 4, we obtainUsing Lemma 2,
Definition 5. The vertex semitotal network of a network is obtained by taking a vertex corresponding to each edge of and then by joining these new vertices to the end points of their corresponding edges.
Theorem 9. For all , the complexity of the vertex semitotal network is given as
Proof. Consider the vertex semitotal network with and , see Figure 9 for its general formation.
Applying Lemma 1, we havewhere and represent the degree and adjacency matrices of , respectively.By using Lemma 4, we obtainUsing Lemma 2,
Definition 6. The edge semitotal network of the network is obtained by inserting a vertex on each edge of and then by joining those new vertices that share common edges in the network .
Theorem 10. For all , the complexity of the edge-semitotal network is given as
Proof. Consider the network with and , see the general form in Figure 10.
Applying Lemma 1, we havewhere and represent the degree and adjacency matrices of , respectively.By using Lemma 4, we obtainUsing Lemma 2,
Theorem 11. For all , the complexity of the network is given aswhere represents the rooted product of and .
Proof. Consider the network with and , see the general form in Figure 11.
Applying Lemma 1, we havewhere and represent the degree and adjacency matrices of , respectively.By using Lemma 4, we obtainUsing Lemma 2,
3. Synopsis and 3D Comparison of the Complexities of the Networks
This section comprises a tabular summary and 3D graphical plots and comparison of the closed formulae for the complexities of networks calculated in this article :(i)Table 1 indicates a brief summary of our findings in the form of complexity of various networks and also categorizes the network being planar or nonplanar(ii)Figure 12 shows the individual 3D graphical tends of the values of the complexity of networks discussed in this article, whereas Figure 13 reveals the relative comparison of the values of the complexity of these networks, reveling the golden one to be the dominated layer among all
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4. Conclusion
The complexity, i.e., number of spanning trees, of a network is a purposeful algebraic invariant. The calculation of this measurement gives an important information about the reliability of a network by providing the information of total number of acyclic networks present within. In this article, we have computed the complexity of networks operation such as , , , , , , , , , , and . Mainly, our techniques have been algebraic and involve Chebyshev polynomials and concepts of the matrix theory while calculating our results.
Data Availability
All the data are included within this paper. However, more details of the data are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that they have no conflicts of interest.