COMPLEXITY OF SOME GRAPHS GENERATED BY SQUARE

unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract. Complexity plays a vital and significant role when designing communication networks (graphs). The more quality and perfect the network, the greater the number of trees spanning this network, which leads to greater possibilities of connection between two vertices, and this ensures good rigidity and resistance. In this work, we present nine network designs created by a square of different average degree 4, 6 and 8, then we deduce a simpler and evident formula expressing the number of spanning trees of these networks using some basic properties of orthogonal polynomials, block matrix analysis technique, and recurrence relations. In addition, we compute the entropy of each network and determine the best by comparing these designs using network entropy. Finally, we compare the entropy of spanning trees on our networks with other triangle and Apollonian networks and observe the entropy of our networks, which is the highest among the triangle and Apollonian networks studied.


INTRODUCTION
In crowded places and occasions telecommunications companies facing significant problems, when someone makes contact with another person and the lines are busy. So another empty communication channel must be provided in order for the communication to take place.
The good company try to increase the number of communication channels, these demands to calculate the number of different tracks (the total number of spanning trees) available within each network and choosing the best design to grantee this increasing.
A graph is a formal mathematical illustration of a network since any network can be modeled by a graph G where nodes are represented by vertices V (G) and links are represented by edges E(G). Let |E(G)| be the cardinality of E(G) and |V (G)| be the cardinality of V (G). We deal with finite and undirected with multiple edges and loops permitted graphs. The degree of a vertex x ∈ V (G) is the number of edges incident with the vertex, while the average degree of a graph is applied to measure the number of edges compared to the number of vertices which calculates by dividing the summation of all vertex degrees by the total number of nodes.
A spanning tree of any graph is a communication subgraph that guarantees connectivity between all vertices of the original graph with a minimum number of edges. In other words, a spanning tree ensures the existence and uniqueness of a connection between any pair of vertices. The number of spanning trees τ(G) is equal to the total number of various spanning subgraphs of G that are trees, this quantity is also known as complexity τ(G) of G.
Many new graphs can be generated from a given pair of graphs using graph operations [1]. There are several methods of finding this number. The celebrated matrix tree theorem of Kirchhoff [2] tells us that: the complexity τ(G) of a graph G is equal any cofactor of Laplace matrix L(G) = D(G) − A(G), where D(G) is the diagonal matrix of vertex degrees of G and A(G) is the adjacency matrix of G . τ(G ) can also be calculated from the eigenvalues of the Kirchhoff matrix H. Let µ 1 ≥ µ 2 ≥ · · · ≥ µ n (= 0) denote the eigenvalues of H matrix.
Kelmans and Chenlnokov [3], have shown that (1) τ(G) = 1 n After that Temperley [4] , has shown that: where J is the n × n matrix all of whose elements are unity.
From Temperley's Equation ( 2), it is easy to prove the following lemma.
Lemma 1.1. Let G be a graph with n vertices and D, A are the degree and adjacency matrices, respectively, of G, the complement of G. Then, The senior advantage of the formula ( 3) is that it directly expresses τ(G ) as a determinant rather than in terms of co-factors or eigenvalues.
For some special classes of graphs, there are simple closed formulas that make it much easier to calculate and determine the number of corresponding spanning trees, especially when these numbers are very large. Cayley showed that a complete graph K n has n n−2 , n ≥ 2 spanning trees [2]. Another result is due to Sedlacek [5], he derived a formula for the wheel with n+1 vertices, W n+1 , has a number of spanning trees ( 3+ 2 ) n − 2, n ≥ 2 . Recently, several closed formulas have been published for counting and maximizing the number of spanning trees for some families of graphs ( see, [6][7][8][9][10][11]).

BASIC PROOF TOOLS
There exists a powerful relation concerning orthogonal polynomials, especially the Chebyshev polynomials of the first and second kinds and determinants that we use in our computations. For positive integer n, the Chebyshev polynomials of the first kind are defined by [12]: For positive integer m, the Chebyshev polynomials of the first kind are defined by: T n (x) = cos(n arccos x) .
The Chebyshev polynomials of the second kind are defined by U n−1 (x) = 1 n d dx T n (x) = sin(n arccos x) sin(arccos x) .
The Chebyshev polynomials of the second kind satisfy the recursion relation U n (x) − 2xU n−1 (x) +U n−2 (x) = 0 . (6) Let A n (x) be n × n matrix such that: From this recursion relation and by expanding det A n (x), one obtains: Using standard methods for solving the recursion ( 6), one obtains the explicit formula Lemma 2.3. Let P, Q and R be matrices of dimension n × n, then Proof. Using the properties of determinants and matrix row and column operations [17] yields:

COMPLEXITY OF SOME FAMILIES OF GRAPHS GENERATED BY A SQUARE
Suppose we have four transmission sources in a broadcasting network, and we want to reach all the nodes in that network without closing that circuit linking those nodes. Therefore, we are going to study this problem in the presence of various models linking these nodes to determine which of these models are the best to connect all nodes.
3.1. Complexity of some families of graphs generated by a square with average degree 4.
Theorem 3.1. For n ≥ 1, the number of the spanning trees of the family of graphs Θ n is given by: Proof. The Kirchhoff matrix associated with the graph Θ n is, Consider the family of graphs Θ n , generated by a square, and constructed as shown in Fig. 1, according to the construction, the number of vertices in the graph Θ n are |V (Θ n )| = 4n + 1 and edges |E(Θ n )| = 8n, n = 1, 2, · · · . When n is large, the average degree of Θ n is 4.

Its characteristic equation is r
The general solution of the Recurrence Relation ( 14 ), is a n = α (3 + 2 Using the initial conditions, we get: . Therefore: From Eq.( 12), Eq.( 13) and Eq.( 15) in Eq.( 10), we get: Consider the family of graphs When n is large, the average degree of ∆ n is 4 .
Theorem 3.2. For n ≥ 1, the number of the spanning trees of the graph ∆ n = C 4 × P n is given by Proof. Applying Lemma (1.1) and Lemma( 2.3), we have: From Lemma (2.1), we obtain: Using the induction and the properties of determinants, we obtain the value From Eq. (17) Consider the family of graphs Λ n , generated by a square, and constructed starting with a square and finding its line graph [1], one gets an inner square. Repeating this process to the new interior square given the graph shown in Fig. 3, the graph Λ n has the number of vertices |V (Λ n )| = 4n and the edges |E(Λ n )| = 8n − 4, n = 2, 3, · · · . When n is large, the average degree of Λ n is 4.
Theorem 3.3. For n ≥ 2, the number of the spanning trees of the graph Λ n is given by Proof. Applying Lemma 1.1, we have: Using the properties of the determinants, we arrive at: Using the initial conditions det(L − M) = 400 , 4624 to n = 3, 4, respectively, we get From Eq. (21) and Eq.( 23) in Eq. (37), we get: Consider the family of graphs It is notice that, when n is large, the average degree of Ψ n is 4.  Proof. Applying Lemma 1.1, we have: Applying Lemma 2.3, with the matrix R = J where J is the unit matrix, we obtain: Applying Lemma (2.2), with x = 2, we get: τ(Ψ n ) = 1 (4n) 2 ( n 2 n ) 2 × 2 n+1 n 2 × 2 n+1 = n 2 2 4n−2 .  Theorem 3.5. For n ≥ 1, the number of the spanning trees of the graph Γ n is given by:

3.2.
Complexity of some families of graphs generated by a square with average degree 6.
Consider the family of graphs ϒ n , generated by a square, and constructed as shown in Fig. 6.
u n v n w n t n FIGURE 6. The graph ϒ n . Theorem 3.6. For n ≥ 1, the number of the spanning trees of the family of graphs ϒ n is given by: τ( ϒ n ) = n 2 (2n + 4) 3 n−2 2 6n−5 .
Consider the family of graphs Π n , generated by a square, and constructed as shown in Fig. 7.
Let a n = b n + d 2 Using the initial conditions det(L − M) = 196 , 6400 at n = 2, 3, respectively, we have: Solving these equations, we get α = 1 14 . Therefore Consider the family of graphs Φ n generated by a square and constructed as shown in to the construction. Note that for n = 2 the graph Φ n is isomorphic with the graph ϒ n . When n is large , the average degree of Φ n is 6.
Theorem 3.8. For n ≥ 3, the number of the spanning trees of the graph Φ n is given by: Proof. let us applying Lemma 1.1, we have: (34)    Consequently, we have the following homogeneous recurrence relation (35) a n+2 = 4 a n+1 − a n .
3.3. Complexity of a graph generated by a square with average degree 8.

ENTROPY OF OUR NETWORKS
Because the complexity of a network τ(G) increases exponentially with the number of vertices, there exists a constant ρ(G), called the entropy of spanning trees [15], described by this relation: (40) ρ(G) = lim n→∞ ln τ(G) |V (G)| .
By calculating the entropy of graphs (networks), we hope to determine the best one. The entropy of spanning trees of a network is a quantitative measure of the number of spanning trees to evaluate the goodness and the resistance of a network and to describe its structure. The most goodness and resistance network is the network that has the highest spanning-tree entropy.
According to the definition of the entropy of spanning trees of a network, the bigger the entropy value, the more the number of spanning trees, so there are more possibilities of connections between two vertices.   Comparing the value of entropy in our design networks (graphs), we have found that: (i) Of all our different design networks (graphs) of average degree 4 , the entropy of the graphs ∆ n and Θ n is the largest.
(ii) Among all our different design networks (graphs) of average degree 6 , the entropy of the graph Φ n is the largest.
(iii) The entropy of the graphs ∆ n and Θ n of average degree 4 is larger than the entropy of all graphs of average degrees 6 and 8.
Now we compare the value of entropy in our design networks (graphs) with other networks: In 2013, Zhang et all [16] proved that the Apollonian graph of average degree 6 has entropy 1.354 . In 2019, Daoud [17] introduced some networks generated by a triangle with average degree 4 and 6, and proved that the graph Y n of average degree 6 has entropy 1.514280 . It is clear that the entropy of our studied networks Π n and Φ n with the same average degree 6 is larger than the entropy of the Apollonian graph, and the graphs generated by a triangle with the same average degree 6. Finally, the entropy of spanning trees of our studied (graphs) networks Θ n and ∆ n of average degree 4 are the highest among these networks.

CONCLUSIONS
The complexity which is the number of spanning trees in the networks is a significant invariant. The enumerating of this number is not only helpful from a combinatorial standpoint, but it is also an important measure of the network reliability and electrical circuit design. In this paper, we introduce nine network designs that are created by a square of average degree 4, 6 and 8 , then we obtain a simple and evident expression for the number of spanning trees of these networks using some basic properties of orthogonal polynomials, block matrix analysis technique, and recurrence relations. Finally, we compute the entropy of these networks and compare the entropy of spanning trees on our networks with the other triangle and Apollonian networks. We deduce that the entropy of our networks is the highest among the studied triangle and Apollonian networks and the networks Θ n and ∆ n of average degree 4 are the highest among these networks.