Analysis of Modified Fifth Degree Chordal Rings

Implementation of new telecommunications services has always been associated with the need to ensure network efficiency required to implement these services. Network efficiency can be described by a number of parameters such as: network bandwidth, propagation time, quality, reliability and fault tolerance. More and better performance, and thus network efficiency is achieved mainly by using more and more advanced technical and technological solutions. There were milestones solutions such as the use of coaxial transmission cables, optical fibers, and various techniques of multiplication like TDM or WDM (Newton, 1996). Significant impact on the way to deliver services had wireless transmission, which has found widespread use in communication networks since the end of last century.

The application of chordal rings in computer systems (Mans, 1999), TDM networks (communication between distributed switching modules) (Bujnowski, 2003), core optical networks (Freire & da Silva, 1999, 2001a, 2001bLiestman et al., 1998;Narayanan & Opatrny, 1999;Narayanan et al., 2001), and optical access networks (Pedersen, 2005;Pedersen et al., 2004aPedersen et al., , 2004bPedersen et al., , 2005Bujnowski et al., 2003) has been analyzed. The authors of this publication, in their earlier works on modeling of telecommunication and computer networks, present an analysis of chordal rings (Bujnowski et al. 2004a(Bujnowski et al. , 2004b(Bujnowski et al. , 2005. In the beginning the general definition of chordal ring will be giving. Definition 1. A chordal ring is a ring with additional edges called chords. A chordal ring is defined by the pair (p, Q), where p denotes the number of nodes of the ring and Q denotes the set of chord lengths Q  {1, 2, ..., p/2}. Since it is a ring, every node is connected to exactly two other nodes (i.e. assume a numbering of the nodes 1,2,…, p -then node i is connected to node i-1 and i+1 (mod p). Node 0 is connected to p and 1). Each chord of length q  Q connects every two nodes of the ring that are at distance q. The chordal ring will be further denoted as G(p; 1, q 1 , ..., q i ), q 1  ...  q i . In general, the degree of chordal rings is 2i, unless there is a chord of length p/2. In this case p should be even and rings' degree is 2i -1 (Gavoille, n.d).
In the papers (Bujnowski et al., 2008a(Bujnowski et al., , 2009b(Bujnowski et al., , 2010Dubalski et al., 2007Pedersen et al., 2009)  and sixth degree chordal rings and modified graphs of these types. These topologies are the subject of many publications of the researchers from Putra University (Farah et al, 2008(Farah et al, , 2010a(Farah et al, , 2010bAzura et al., 2008Azura et al., , 2010Farah et al. 2010Farah et al. , 2011. In this publication the survey of the chordal rings consisting of fifth degree nodes (Fig. 1) will be presented. Until now this type of the regular structures is not widely examined, so authors decided to focus on it (Dubalski, 2010).
where v i means the number of the node, d min minimal distance (number of edges) between ith and j-th node.
Definition 3. The average path length d av between all pairs of nodes is defined by the formula: where d min (v i , v j ) is the minimal number of edges between a source node v i and every other chosen node v j ,, and p denotes the number of nodes.
A Reference Graph (a virtual example is shown in Fig. 2) can be determined, which presents a reference for all regular graphs of degree 5. It represents lower bounds for average distance and diameter for all these graphs, but since it is a "virtual graph" these bounds may not always be achievable.
The Reference Graph possesses parameters as follows: 1. The number of nodes p dr in d-th layer is determined by formula: 5. This graph is symmetrical, its all parameters are equal regardless from which node they are calculated.

Fig. 2. General diagram of virtual infinite Reference Graph
Only one Reference Graph fifth nodal degree exists in reality, it is the complete graph consisting of 6 nodes.
Two other reference graphs, named as Ideal and Optimal graphs, are also useful for determining average distance and diameter of the chordal rings. They provide theoretical values, which in the following will be compared to values obtained in the real graphs. As for the reference graph mentioned above, the optimal and ideal graphs do not always exist.
In order to determine parameters of the theoretical calculated reference topologies of chordal rings two types of these structures were defined. The first one is called the ideal graph and the second one -optimal graph. In fact these graphs exist only in particular cases, but they are useful as reference models for evaluation expected parameters of tested graphs.
Definition 4. The ideal chordal ring with degree D(G) is the regular graph with total number of nodes p i given by the formula: where p d means the number of nodes that belong to the d-th layer (the layer is the subset of nodes that are at a distance d from the source node), while p D(G) denotes the number of the remaining nodes which appear in the last layer. For ideal rings, for every n and m < D(G) nm p p  = Ø. If for certain D(G) the subset p D(G) of chordal ring reaches the maximal possible value, then such a ring is called the optimal ring (optimal graph).
For ideal chordal ring the average path length d avi is expressed as: whereas for the optimal graph the average path length d avo is equal to: where d -layer number, p d -number of nodes in d-th layer, p o -number of nodes in optimal graph.
Optimal graphs were used to calculate the formulas describing parameters of each type of analyzed chordal ring, whereas ideal rings were served to compare calculated theoretically and obtained in reality parameters of analyzed structures.
The basic topology of fifth degree chordal rings in Fig. 3 is shown. The definition, short presentation and author's consideration concerned of this structure are given below. Definition 5. The basic chordal ring fifth nodal degree called CHR5 is an undirected graph, based on a cycle with additional connections (chords). It is denoted by CHR5(p; q 1 ,q 2 ) where p must be even and means number of nodes creating the ring, chord length q 1 > p/2 is odd, even too, chord length q 2 is equal to p/2. The values of p and q 1 must be prime each other (Bujnowski, 2011).
In order to calculate the diameters and average path lengths appearing in optimal graphs it is necessary to evaluate the maximal number of nodes appearing in each layer. If d > 1 the power of these sets is described by formula: Using the formula given above, the total number of nodes p o in the optimal graph with diameter D(G) can be calculated (D(G) > 1): The total number of nodes in optimal graphs versus its diameter is shown in  Table 2. Total numbers of nodes forming optimal graphs versus diameter The average path length in optimal graphs is given by formula:   32 2 8() 6() 2() 3 Only one optimal graph exists in reality. It is the complete graph which possesses 6 nodes, but the ideal chordal rings can be found. Whereas it founded two groups of ideal graphs consisting of p nodes, which can be described by formulas given below.
The graphs belonging to the first group are defined as follows: The graphs belonging to the second group are described as follows: If D(G) > 2 then: So when the number of nodes is equal to p i then The lengths of chords used to construct ideal graphs can be calculated using formulas: When the number of nodes creating chordal ring is given by equation: 22 11 2 2 1 2 2 1 4(9 9 2) 6(6 10 4) (1, 2, , ) 2() 1 1 2 4() 2() 3 4(9 9 2) 3 4() 6() 3 6(6 10 4) 3 The average path length of all these graphs is described by formula: Unfortunately the parameters of CHR5 graphs are considerably different of Reference Graph parameters, what is shown in fig. 4 and 5 given above.
It follows from the difference of number of nodes appearing in successive layers and thus the difference of total number of nodes appearing in dependence of its diameter as well.
www.intechopen.com The aim of authors of this publication was to find structures possessing basic parameters which values would be closer to reference graph parameters.

Analysis of modified graphs fifth degree
The authors prepared two programs which were used to make it possible to examine the analysed graphs -"Program Graph Finder" and "Find the best distribution of nodes in the layers". The first one -"Program Graph Finder" was used in the first stage of analysis for quite simple topologies, the second one "Find the best distribution of nodes in the layers"for more complicated structures, when the number of variables describing the way of connections is greater than 4.
The real values of parameters of modified chordal rings were calculated using these programs and compared to those obtained in a theoretical way.
In the following sections, an analysis of 15 different regular structures based on chordal rings is presented. Each of the type of graphs is defined, examples are given, the distribution of nodes in different layers is analyzed, and the ideal and optimal graphs are compared to real graphs. Also, basing on the analysis of nodes in different layers, the average distance and diameter can be calculated as a function of the number of nodes.
The graphs are divided into 3 groups, each consisting of 5 types of graphs. The first group of graphs needs to have a number of nodes divisible by two, and the second group of graphs a number of nodes divisible by 4. The third group of graphs also has a number of nodes divisible by 4, but for these no mathematical expressions of node distribution (and thus the average distance and diameter) were found.

First group of chordal rings
As previously mentioned, for each type of graph we present: Expressions for key parameters  Comparisons of parameters for real and theoretical graphs.

Graph CHR5_a.
Definition 6. The modified fifth degree chordal rings called CHR5_a ( Fig. 6) is denoted by CHR5_a(p; q 1 ,q 2 ), where p is even and means number of nodes; q 1 , q 2 are chords. Chords q 1 and q 2 are odd and < p/2. Chord q 1 generates a Hamiltonian cycle. whereas q 2 is odd too and < p/2. Each even node i 2k is connected to five other nodes: The values of p and q 1 must be prime each other (this ensures that the Hamiltonian cycle is created).  In The general expression has the following form: 2 1 (7 4 (mod 2)) 2 The total number of nodes p o forming an optimal graph which possesses diameter D(G) is expressed as: which confirms the results obtained by constructing the possible graphs, as shown in Table 4.  Table 4. Diameters and total numbers of nodes in virtual, optimal graphs The average path length in optimal graphs can be calculated as:  Definition 6. The modified fifth degree chordal ring called CHR5_b is denoted by CHR5_b(p; q 1 ,q 2, q 3 ) where p is even and means number of nodes; q 1 , q 2 , q 3 are chords, where chord q 1 and q 2 possess even lengths, whereas the length of q 3 is odd. The values of p and q 1 , q 2, q 3 must be lower than p/2. Each even node i 2k is connected to five other nodes:   Table 5. Maximal number of nodes in the layers When d is bigger than 2, the maximal number of nodes which can appear in the successive layers is described by: The total number of nodes p o in the optimal graph with diameter D(G) > 1 is given by: This was also confirmed by constructing the possible graphs. These results can be seen in Table 6.  Definition 7. The modified fifth degree chordal ring called CHR5_c is denoted by CHR5_c(p; q 1 ,q 2, q 3 ), where p is even and means number of nodes; q 1 , q 2 , q 3 are chords, all chords possess odd lengths less then p/2. The values of p and q 1 , q 2, q 3 must be prime each other. Each even node i 2k is connected to five other nodes: . 10 shows an example of CHR5_c. In Table 8 the numbers of nodes in the layers of an optimal graph is shown.  In the case when d -number of layer is bigger than 2 the number of nodes in the layers can be described by the following expression: 32 26 7 55 3 0 33 The total number of nodes p o in the optimal graph with diameter D(G) > 1 is given by: In table 9 the total number of nodes in virtual, optimal graphs, as described by the above expression, is shown.  Table 9. Total numbers of nodes in optimal graphs The average path length in optimal graphs can be calculated using the expression: www.intechopen.com  11 shows a comparison of diameter and the average path length between theoretical and real graphs. All graphs of this type are symmetrical, but they couldn't find any ideal graph.

Graph CHR5_d.
Definition 8. The modified fifth degree chordal ring called CHR5_d is denoted by CHR5_d(p; q 1 ,q 2, p/2), where p means the number of nodes and is positive and even; q 1 , q 2 are chords which possess odd lengths less then p/2. The values of p and q 1 , q 2 must be prime each other. Each even node i 2k is connected to five other nodes:   When the number of layer d is bigger than 1, the number of nodes in the layers can be described by the following expression: The total number of nodes p o in the optimal graph depending on the diameter is given by: In Table 11 the total number of nodes in virtual optimal graphs described by formula given above is shown.  Table 11. Total numbers of nodes forming optimal graphs versus diameter The average path length in optimal graphs can be calculated using the expressions:   All this type of chordal rings are symmetrical. Ideal graphs are only for the cases where the number of nodes is divisible by 4. Examples are given in Table 12 Graph CHR5_e.
Definition 9. The modified fifth degree chordal ring called CHR5_e is denoted by CHR5_e(p; q 1 ,q 2, p/2), where p means the number of nodes and is positive and even; q 1 , q 2 are chords which possess even lengths less then p/2. The values of p/2 and q 1 , q 2 must be prime each other. Each even node i 2k is connected to five other nodes:  Table 13 the number of nodes appearing in the successive layers of optimal graphs is shown. It should be observed that there are two different number of nodes in layers, depending on the total number of nodes are they divisible by 4 or not.
The total number of nodes p o in the optimal graph depending on the diameter is given by: In Table 14 the total number of nodes in virtual optimal graphs described by the above expressions is shown.  Table 14. Total numbers of nodes forming optimal graphs versus diameter The average path length in optimal graphs can be calculated using this expression:   Some but not all of these graphs are symmetric, and as illustrated in Fig. 15  To sum up, in the first group of analyzed graphs the best parameters have CHR5_b graphs. They possess minimal diameter and average path length in comparison to the other analyzed chordal rings, and the parameters of the real graphs are close or equal to parameters of ideal graphs. Additionally, most of the best graphs are symmetric, what is also an advantage for the application in real networks.

Second group of analyzed graphs
The chordal rings consisting of 4i nodes (i = 2, 3, 4, …,) belong to this group. These topologies are often more complicated, since they are less symmetric. Basing on patterns for ideal and optimal graphs it is possible to derive expressions for average distance and diameter for all of the different topologies in this group of graphs. Definition 10. The modified fifth degree chordal ring called CHR5_f is denoted by CHR5_f(p; q 1 ,q 2 ,q 3 , p/2), where p means the number of nodes and is positive and divisible by 4; q 1 , q 2 , q 3 , are chords which possess even lengths less then p/2. The values of p/4 and q 1 , q 2, q 3 must be prime each other. Each even node i 2k is connected to five other nodes: An example is shown in Fig. 17. This structure is more complicated since the number of nodes appearing in successive layers depends on whether the total number of nodes is divisible by 8 or not, and also on whether it seen from odd or even node number in the graph. This creates multiple cases, which also complicates deriving the basic parameters. Table 16 shows the experimentally obtained results, which are the basis for further analysis.

62
In Table 17 the total number of nodes in optimal graphs given by above expression is shown.

D(G)
An example is shown in Fig. 21, and the number of nodes in the layers of an optimal graph is shown in tables 21 and 22.   The number of nodes in the layers, as shown in Tables 21 and 22, In Table 23 the total number of nodes in optimal graphs described by expression given above is shown.

Graph CHR5_i.
Definition 13. The modified fifth degree chordal ring called CHR5_i is denoted by CHR5_i(p; q 1 ,q 2 ,q 3 ,q 4 ), where p means the number of nodes and is positive and divisible by 4; q 1, q 2 , q 3 , q 4 are chords which possess: q 1, odd length and q 2 , q 3 , q 4 -even lengths less then p/2. Each node is connected to five other nodes. Even node i 2k is connected to i 2k-1 , i 2k+1 , i 2k+q1(mod p) , i 2k-q1(mod p) and i 2k+q2(mod p) , while odd node i (2k+1)=1(mod4) is connected to i 2k , i 2k+2 , i 2k+1-q2(mod p) , i 2k+1+q3(mod p) , i 2k+1-q3(mod p) and odd node i (2k+1)=3(mod4) is connected to i 2k , i 2k+2 , i 2k+1-q2(mod p) , i 2k+1+q4(mod p) , i 2k+1-q4(mod p) . An example of a CHR5_i is shown in Fig. 23. The distribution of nodes in the layers depends on whether the graph is seen from odd or even node in Table 24   The distribution of nodes in the layers can be described by the expression: The total number of nodes in optimal graphs calculated depending on the source node number, given in  For graphs with less than 72 nodes it is possible to find real graphs with parameters close to those of ideal graphs. However the difference becomes bigger for larger graphs. The differences seem to come from the different path lengths calculated from odd and even nodes. The ideal graphs found are shown in Graph CHR5_j.
Among all graphs belonging to the second group the best parameters (minimal diameter and minimal average path length given the number of nodes) were found in CHR5_i but other in minimal degree are slightly different it (especially from CHR5_j). In Fig. 27 the comparison of the second group of graphs is shown.

Analysed graphs -Third group
There are a number of other topologies, for which we have not found any nice expressions for the distribution of nodes in layers, and thus no expressions of the average distance and diameter could be derived. Due to the good basic parameters the topologies have been described, but further research is needed in order to provide more precise descriptions.
An example is shown in Fig. 28.  (20; 4,8,4,8,2,6) In Table 29 the distribution of nodes in layers is shown, based on observations of all graphs. Using the results shown in Table 29, the counted total number of nodes in virtual optimal graphs is presented in table 30. 1  2  3  4  5  6  7  8  p o  6  26  106  390  1285  3805  10138  24472   Table 30. Total numbers of nodes forming optimal graphs versus diameter

D(G)
In Fig. 29 comparison of diameter and average path length of theoretical and real graphs CHR5_k is shown.
In Table 33 the distribution of nodes in layers is shown. The total number of nodes in optimal graphs, calculated based on these results, are shown in Table 34.   Table 34. Total numbers of nodes forming optimal graphs versus diameter In Fig. 34 comparison of diameter and average path length of theoretical and real graphs CHR5_m is shown.
An example of a CHR5_n graph is shown in Fig. 34. Fig. 34. Example of modified chordal ring CHR5_n(20; 2,6,10,2,6,10) In The total number of nodes in optimal graphs calculated based on the results given in Table  35 is shown in Table 36.  Table 36. Total numbers of nodes in optimal graphs as a function of the diameter In Fig. 36 shows the comparison of diameter and average path length of theoretical and real graphs CHR5_n. Going through all nodes with up to 100 nodes only one ideal graph was found, namely CHR5_n(52;10,6,14,18,26,22) with 52 nodes.

Analysis of obtained results
Based on the obtained results for all 15 groups of graphs presented, the values of minimum diameter and average paths lengths can be compared. Despite the differences found between theoretical and real parameters the comparisons will be based on the theoretical values.
First, Fig. 39 presents the average path lengths as a function of the diameter in the graphs. This does not take into account the number of nodes in the graphs, which vary significantly between the different graphs as can be seen in Table 39 and Fig. 40. It can be seen that for a given number of nodes, CHR5_k has the smallest diameter. In the following section, a comparison of the average path length as a function of the number of nodes is presented. In order to compare graphs with different numbers of nodes, a "Normalized estimator of average path length" (E nav ) is introduced as follows:   The results are shown in Table 40 and Fig. 41.  Fig. 41 it can be seen that he CHR5_k has the relatively shortest average path length.
Not surprisingly, the distributions of the number of nodes in the layers have a great impact on the two basic parameters. This can be seen also from expressions (7) and (9). The difference in the distribution for all graphs is shown in Fig. 42.  In the charts given below the comparison of the maximum number of nodes appearing in the layers of all analyzed graphs is presented.
On the basis of Fig. 42 and Fig. 43 it seems to be sufficient to analyze the distributions of numbers occurring in the first few layers to select a graph having the best basic parameters. This obviously reduces the time and effort for comparisons. The results again confirm that CHR5_k seems to be superior in terms of these basic parameters. In order to make an objective assessment of the CHR5_k parameters, they were compared to the parameters of the Reference Graph as previously described. Table 41 and Fig. 44 show the distribution of nodes in different layers of these two graphs, as a function of their diameters.   Fig. 45 a comparison of node numbers in successive layers and total number of nodes as a function of number layer in ideal CHR5_k and Reference Graph is shown.

Conclusions
In this publication the analysis of a different construction of 5 th degree chordal rings was presented. The authors' main aim was to find structures which have the minimal diameter and minimal average of path in respect to number of nodes which create these graphs.
Presented considerations in the paper have rather theoretical nature, without a strict reference to practical applications. It is difficult to imagine a real regular WAN communication network that consists of thousands of nodes. However the interconnection structures connecting thousands of microprocessors, or sensors require the construction of such networks as the regular ones. The main objective function for regular interconnection structures is to minimize the network diameter or average path length. So, this is the main reason why such the structures were analyzed and studied in our paper.
In this regard the program was worked out which allows to calculate the analyzed parameters. It allowed describing virtual reference graphs namely optimal and ideal graphs.
In this way we also found chordal rings which possess the smallest difference regards to average path length and diameter, which Reference Graphs have. They examined many types of structures and concluded that parameters of the real graphs are slightly different of the theoretically calculated graph parameter values. The obtained results became the basis for preparing the general formulas for determining these parameters without the need of simulation. As a side-result of the paper, we have shown that these reference graphs can be used for obtaining fairly good estimation of distance parameters in a simple manner. Additionally, they concluded that it is enough to inspect the maximal number of nodes which can appear in first few layers in aim to choose the best topology.
This publication presents the results of analysis of the modified chordal rings fifth degree. This analysis was carried out for 15 graphs divided into 3 groups. Each of the group included 5 types of graphs. Since the graphs were analyzed are regular graphs odd degree, hence all the graphs have to have an even number of nodes. The nodes number of all graphs belonging to the first group is divisible by two, the second and the third by four (it follows from used method of their construction). For each group of graphs their analysis based on results obtained thanks to the application of testing programs constructed by authors. It made possible to carry out a distribution of maximal number of nodes in layers, to count total number of nodes in virtual, optimal graphs. Based on obtained results, for the first two groups of graphs they found strict mathematical expressions describing the distribution of nodes in layers, the total number of nodes, the average path length for optimal graphs, whilst for the third group such formulas were not found. Additionally the prepared programs allowed us to compare the basic parameters of real and theoretically constructed graphs.
Among the all analyzed graphs, the structures CHR5_f -CHR5_o have the most acceptable basic parameters. Those graphs however have a fundamental limitation: the network should consist of nodes with nodes number has to be divisible by 4. The parameters of these networks in more or less deviate from the parameters of the reference graph (graph ideal), and what's involved, they are usually asymmetrical (depending on the choice of the source node, obtained values differ). Also, computational process is rather complex, and takes long time.
From the point of view of application, according to the authors, the most appropriate structure of the regular network topology are graphs CHR5_b. These chordal rings are symmetrical, their parameters, if are not equal to the parameters of optimal graphs, they are very close to them; they are simple and easy to design and implement. The main limitation is the fact that the number of nodes, creating these networks, has to be even, but this follows from the assumption that the structure has to be regular, and the degree of nodes is five. Fig.  47 shows the comparison of the best two structures. As the justification of this last conclusion the presented above diagrams can be used. In Fig.  47 we show the comparison of basic parameters of chordal rings CHR5_b and the best graph -CHR5_k. It can be observed that in up to 84 nodes, theoretically calculated parameters of both types of rings are identical, and up to 224 nodes -not much different from each another. Thus, taking into account the advantages of CHR5_b structure described before, it should be used to construct regular networks possessing not a huge number of nodes or in a large network consisting of a few identical regular structures.
Future work can focus on both theoretical and practical aspects. For the theoretical aspects, it would be a big help to find more precise and reachable bounds. This would make it possible to assess graph types, and to know how close to optimal they are. Moreover, a more thorough study of how precise distance estimates can be given using ideal and optimal graph would be interesting. Such a study could also cover other types of graphs. Another direction for further research would be to study new groups of graphs.
More practical aspects could deal with analyzing how well the good theoretical properties translate into good network properties. This could be done through simulation of different network configuration and traffic scenarios, and/or by studying how feasible the graphs are for assignment of physical, optical or logical links. In order to demonstrate that the topologies are useful in real-world settings, case studies would be a good place to start.