Abstract
The connected and acyclic components contained in a network are identified by the computation of its complexity, where complexity of a network refers to the total number of spanning trees present within. The article in hand deals with the enumeration of the complexity of various networks’ operations such as sum (, , ), product (, , , ), difference , and the conjunction of with . All our computations have been concluded by implementation of the methods of linear algebra and matrix theory. Our derivations will also be highlighted with the assistance of 3D plots at the end of this article.
1. Introduction
Only simple network shall be dealt with throughout the paper. One of the most useful algebraic invariants is the complexity, i.e., number of spanning trees in a network admitting roots in combinatorics, algebraic graph theory, and networking. It is prominently linked with network engineering and particular branches of computer sciences that deal in the security designs specifically. Realistically, concreteness and precision in a network are based on the number of spanning trees it possesses. This indicates that complexity is an identifier for the quality of a network. Certain applications of complexity in different fields of mathematics and physics can be observed in [1–4]. For instance, we are living in an era of networking. The tools similar to complexity ensure the robustness and accuracy in a network so that one can obtain interruption free signals, since the complexity is an identifier of the number of connected and acyclic pathways present in a network, where every such pathway contains all junctions or vertices present in a network. So, this invariant helps in the enhancement of robustness of wireless sensor networks (WSNs) and other similar mobile networks by relating the total number of spanning trees present within. Another application of complexity can be observed in the security design of a sensitive area of a building. Say there are several secured chambers, and there are legitimate passages only to reach to those chambers. One legitimate passage can be identified by a unique pathway. That is, no cyclic pathway is allowed from one chamber to another. A programming-based software application will ensure if a visitor follows a legitimate passage or not through acyclic pathway mechanism, whereas such unique acyclic pathway is termed as complexity of the network.
1.1. Definitions and Preliminaries
The following lemma is a direct derivation of Temperley’s equation mentioned previously.
Lemma 1 (see [5]). Let be order network; then,where .
The above expression is more useful as it represents the complexity of as the determinant of a particular matrix, rather than involving its eigenvalues. The eigenvalues based process is relatively difficult and complex.
The solution of the following iterative expression defines the first kind of Chebyshev polynomials.
The standard solution of (2) gives
The solution of the following iterative expression defines the second kind of Chebyshev polynomials.
The standard solution of (4) gives
Identity (4) is valid excluding [6]. The determinants are closely related to both and kind Chebyshev Polynomials.where and are non-singular matrices.
Lemma 2 (see [7, 8]). (i), , where(i), , , where(i), , where(i), , , where
Lemma 3 (see [9]). and , , whereis ancirculant matrix given as
Lemma 4 (see [10]). Let , , , andbe the block matrices of orders, , , and, respectively. Then,
Lemma 5 (see [11]). For , let us consider a circulant matrix given as
We shall also provide a few definitions [12, 13] in Section 3 before a certain result, where necessary. Throughout the article, represents the complement of the network .
1.2. Main Contributions
In the present article, we will mainly compute the closed formulae for the complexity of various generalized operations on graphs such as sum (, , ), product (, , , ), difference , and the conjunction of with . Furthermore, all our computations have been concluded by implementation of the methods of linear algebra and matrix theory.
1.3. Main Structure
The main structure of this article is as follows:1.Section 1 comprises the introduction and preliminaries of our main work.2.Section 2 contains the salient work related to our derivations.3.Section 3 consists of the main derivations we have obtained in the form of the complexities of various networks’ operations.4.Section 4 contains a brief summary and graphical illustrations of our work.5.Section 5 gives the conclusion and also tells about the future work related to this paper.
2. Related Work
If we talk about the closed formulae for the complexity of an infinite family of networks, we shall not be able to locate any such generalized result. Although it is still possible to derive the new closed formulae of the complexity of classes of networks having order , where is sufficiently large, it is useful to obtain this invariant for the networks of finite order for the values as we increase the order of a network. If we look into the historical development of this concept, the calculation of the complexity of the complete network as is the foremost concept that appeared in [14]. The second prominent result in this regard is the complexity of the complete bipartite network which is again derived by Cayley [14] as . In [15], the closed formula for the complexity of Mobius ladder has been obtained as for in [15].
The determination of the total spanning trees of a network has recently reappeared as an active topic. Kirchhoff’s matrix tree theorem [16] is a prominent result in this regard. It represents the complexity of a network as the determinant of a random cofactor of its Kirchhoff’s matrix, where, say for a network , indicates its Kirchhoff’s matrix.
A combinatorial method for computing the complexity of a network is with the use of contraction-deletion theorem. As an iterative process for an edge , the complexity of is the sum of and . Here, is the network derived as the result of contraction of in repeatedly until the end points and coincide [17].
In [18], the self-adapted task scheduling strategies in the wireless sensor networks have been designed and analyzed. Wang et al. [19] discussed the ant colony optimization-based location-aware routing for wireless sensor networks. In [20], a pedestrian detection method has been designed and examined based on the genetic algorithm for optimizing XGBoost training parameters. For wireless sensor networks, Wan and Xiong designed and assessed an energy-efficient sleep scheduling mechanism with similarity measure [21]. Lu et al. in [22] explored a finger vein-based personal authentication mechanism for Internet-related security. Furthermore, in [23, 24], some latest work on the enumeration of the complexity of networks can be observed.
3. Main Results
In networking, the characteristic of developing new structures from the existing ones through network operations and studying their various properties always remains active. The present section addresses our main derivations consisting of the closed formulae of the complexity of various networks obtained as the result of network operations.
Theorem 1. For all , the complexity of the network is given by
Proof. Consider the network with and (see the general formation in Figure 1).
Applying Lemma 1, we haveNow on the above determinant, we perform the following operations simultaneously:(i)Adding all columns to .(ii)From C1, we take the number n+5 as common.(iii)Subtracting from all columns.(iv)Expanding along .This yieldsBy using Lemma 4, we getEvaluating and simplifying, we obtain .
Theorem 2. For all , the complexity of the strong product is given by
Proof. Consider the network with and (see the general formation in Figure 2).
Applying Lemma 1, we haveBy using Lemma 4, we getEvaluating the above determinant, we obtain finally .
Theorem 3. For all , the complexity of the homomorphic product is given by
Proof. Consider the network with and (see the general formation in Figure 3).
Applying Lemma 1, we haveBy using Lemma 4, we getEvaluating the above determinant and simplifying, we obtain finally .
Theorem 4. For all , the complexity of the mirror network is given as
Proof. Consider the network with and (see Figure 4).
Applying Lemma 1, we haveBy using Lemma 4, we haveSimplification finally gives .
Theorem 5. For all , the complexity of the cartesian product is given as
Proof. Consider thenetwork with and (see Figure 5).
Applying Lemma 1, we haveBy using Lemma 4, we haveSimplifying, we get .
Corollary 1. For all , the complexity of the symmetric difference is given as
Proof. Since (see Figure 6), .
Theorem 6. For all , the complexity of the strong product is given as
Proof. Consider the network with and (see Figure 7).
Applying Lemma 1, we haveSimplifying, we get .
Theorem 7. For all , the complexity of the symmetric difference is given by
Proof. Consider the network with and (see the general formation in Figure 8).
Applying Lemma 1, we haveBy using Lemma 4, we getEvaluating the determinant and simplifying, we obtain finally .
Theorem 8. For all , the complexity of the network is given by
Proof. Consider the network with and . Applying Lemma 1, we haveBy using Lemma 4, we obtainUsing Lemma 3 and simplifying, we get .
Theorem 9. For all , the complexity of the conjunction is given as
Proof. Consider the network with and (see Figure 9).
Applying Lemma 1, we haveBy using Lemma 4, we haveLemma 3 and simplification finally give
4. Synopsis and the Diagrammatic Comparison of the Complexities of the Networks Obtained
This section consists of a briefing and graphical plots and juxtaposition of the values of complexities of the networks enumerated in this note.
Table 1 indicates a synopsis of our results in the shape of complexities of various networks and also categorically recognizes it being planar or not.
Figure 10 shows the discrete graphical shapes of the values of the complexity of networks obtained here, whereas Figure 11 addresses the relative comparison of the complexities of these networks, revealing the red one to be the dominating layer.
5. Conclusion
One of the meaningful algebraic invariants in networking nowadays is complexity. This invariant provides us the information of the total number of acyclic networks present within the base network, which ultimately ensures the reliability and accuracy in the network. We have enumerated here the complexity of various operations on networks such as , , , , , , , , and . The adopted methods are mainly algebraic and feature Chebyshev polynomials and the matrix theory in the calculations. As future work, we encourage the researchers to obtain the complexities of further generalized operations on networks such as corona product, zig zag product, homomorphic product, join, shadow, conjunction, and disjunction of various classes of networks.
Data Availability
The whole data are included within this article. However, the reader may contact the corresponding author for more details on the data.
Conflicts of Interest
The authors declare that they have no conflicts of interest.