Optimal Graphs in the Enhanced Mesh Networks

+e degree diameter problem explores the biggest graph (in terms of number of nodes) subject to some restrictions on the valency and the diameter of the graph. +e restriction on the valency of the graph does not impose any condition on the number of edges (apart from taking the graph simple), so the resulting graph may be thought of as being embedded in the complete graph. In a generality of the said problem, the graph is taken to be embedded in any connected host graph. In this article, host graph is considered as the enhanced mesh network constructed from the grid network. +is article provides some exact values for the said problem and also gives some bounds for the optimal graphs.


Introduction
All graphs discussed in paper are simple, finite, and undirected. e valency (or degree) of a node (or vertex) in the graph G is the number of edges connected with that node in G. e maximal valency of the graph G is indicated by Δ(G). e distance between two nodes u and v of G is the length of the shortest path between them. e distance between a node x and the set A is defined as d(x, A) � min a∈A d(x, a). For any A ⊆ V(G), let G − A denote the subgraph of G after removing from G all the nodes of A and all the edges incident to at least one node of A. e diameter of the graph G is designated by D and is described as the largest distance between any pair of nodes of the graph G. For a given graph G and natural numbers Δ and D, N G (Δ, D) indicates the number of nodes of largest subgraph of G with given maximal valency Δ and given diameter D.
e topology of a network such as a telecommunications, multiprocessor, or local area network is generally represented by a graph such that stations or processors are represented by vertices (or nodes) and the links or the connection between these networks are represented by edges. ere are many important features in the designing of such networks. One of the important aspects is to put limitation on vertex degree and its diameter. ese two parameters in networks are interpreted as follows. By the degree of a node, it is meant to have the number of connections attached to that node; on the contrary, the diameter shows the largest number of links that must be required to transmit a message between any two nodes. e natural question that arises in this case is "What is then the largest number of nodes in a network with a limited degree and diameter?" If we design the network so that there is no directed edge, then this leads to the Degree/Diameter Problem. More formally, we define it as follows.
Find the largest possible number of vertices N(Δ, D) in a graph of maximum degree Δ and diameter D.
For a thorough survey of the state of the problem, see [1,2]. In the Degree Diameter Problem, only restriction that is imposed on the edges is the maximum degree, so there is considerable freedom in placing edges so as to avoid violating the diameter constraint. In this way, the resulting graph may be thought of as being embedded in the complete graph. In this case, the complete graph is acting as the host graph.
A generalization of the Degree Diameter Problem is to consider the graph as embedded in some host graph, not necessarily the complete graph. is problem becomes more interesting when the host graph is considered as a graph obtained from some network. e problem was first posed by Dekker et al. [7] in the following form.
Given connected undirected host graph G, an upper bound Δ for the maximum degree, and an upper bound D for the diameter find the largest connected subgraph of maximum degree ≤ Δ and diameter ≤ D.
In [3][4][5][6], the degree diameter problem of honeycomb network, triangular network, oxide network, and silicate network has been explored. In [7], the largest subgraph N G (Δ, D) has been determined with multidimensional hexagonal grid as the host graph. In [8][9][10][11][12][13][14][15], some extremal properties of graph networks are discussed. Mesh networks can relay messages using either a flooding technique or a routing technique. With routing, the message is propagated along a path by hopping from node to node until it reaches its destination. To ensure that all its paths are available, the network must allow for continuous connections and must reconfigure itself around broken paths, using self-healing algorithms such as Shortest Path Bridging. Self-healing allows a routing-based network to operate when a node breaks down or when a connection becomes unreliable. As a result, the network is typically quite reliable as there is often more than one path between a source and a destination in the network. Although mostly used in wireless situations, this concept can also apply to wired networks and to software interaction. In this work, we have extended this study to the enhanced mesh network.
e Cartesian product S□T of the graphs S and T is the graph with node set V(S) × V(T), where two nodes (x 1 , y 1 ) and (x 2 , y 2 ) are adjacent if and only if either x 1 � x 2 and y 1 y 2 ∈ E(T) or y 1 � y 2 and x 1 x 2 ∈ E(S).
Let P m and P n are two paths having nodes m and n, respectively. e graph of the grid network P m □P n is obtained by their Cartesian product. e grid graph network P m □P n has mn nodes and 2mn − m − n edges. e graph of the two-dimensional infinite grid network is denoted by P ∞ □P ∞ .
Let G m,n be the graph of the enhanced mesh obtained from P m □P n by replacing its each 4 − cycle by a wheel W 5 , the hub of the wheel being a new vertex (see Figure 1). In the graph G m,n , each 4-cycle of P m □P n is divided into four triangles where every node lie on some triangle.
Let us define the edges of the wheel W 5 as follows: Since P m □P n has (m − 1)(n − 1) squares (4-cycles). erefore, G m,n has mn + (m − 1)(n − 1) � 2mn − m − n + 1 nodes. In G m,n , the edges adjacent to the hub vertex in each W 5 are disjoint; therefore, G m,n has 2mn − m − n + 4(m − 1) (n − 1) � 6mn − 5m − 5n + 4 edges. Furthermore, we define that any two wheel graphs W 5 in G are said to be adjacent if they share an edge.

Methodology
In this paper, we have calculated N G (Δ, D) in the infinite enhanced mesh network. First, we find the induced closed balls G D of diameter D for Δ � 8 whose order is N G (8, D) for D ≥ 2. e order of G D is the upper bound for N G (Δ, D). We find N G (Δ, D) for Δ ≤ 7 by deleting minimum number of vertices of G D . e rest of the paper is ordered as follows. In Section 3, we consider the case for Δ � 8; in Section 4, the case for Δ � 1, 2, 3, 4, 5, 6, 7 is discussed.

Result and Discussion
In this section, the results obtained are discussed. e largest subgraphs are obtained for the enhanced mesh networks for given degree and diameter.
Proof. For even diameter D, let G D be a closed ball having radius D/2 with center as an eight degree node n of G with node set x ∈ V(G): d G (x, n) ≤ (D/2) . To find the nodes of G D , we draw horizontal lines on the nodes of G D and count the nodes on these horizontal lines by adding them from top to bottom. us, for D � 2k, k ∈ N, we have Now, for odd diameter D, let G D be a closed ball having radius (D − 1)/2 and center as the fixed triangle T with node set x ∈ V(G): d G (x, T) ≤ ((D − 1)/2) . en, by counting the nodes on horizontal lines, we have for D � 2k + 1, k ∈ N, V G D � 2 + 3 + 4 + · · · +(2k − 1) + 2k +(2k + 1) +(2k + 2) +(2k + 1) + 2k +(2k − 1) + · · · + 4 + 3 + 2 � 4k 2 + 8k + 2. (3) In Figure 2, the graphs G D for D � 8, 9, 10, and 11 are depicted. e central vertex n of G D for even D is depicted by • and for odd diameter D, the nodes of central triangle T of G D are indicated by a, b, and c and is depicted by •. It is important to note that the distance of any node x from T is defined as d(x, T) � min y∈V(T) d(x, y), where the node y is the nearest node of T from x.
Let L D be the biggest connected subgraph of G with maximal degree Δ � 8.
en, for D � 1, L D is the triangle K 3 ; thus, Hence, we showed the following statement. □ Theorem 1. Let G be the graph of enhanced mesh and D be a positive integer. en, Note that the values in the abovementioned theorem are also trivial upper bounds on N G (Δ, D) for Δ ≤ 7. us, we prove the following corollary.

Largest Subgraphs for Δ ≤ 7
We are interested to find the biggest subgraph of G with given maximal valency Δ and diameter D. Since Δ(G) � 8, it makes perception to consider the cases for Δ ≤ 8. For Δ � 8, Proof. For D � 2k, k ∈ N. Let L and M be the horizontal and vertical lines, respectively, that are passing through the central node n of L D and divide the graph L D into four regions (upper left, upper right, lower left, and lower right), as shown in Figure 3. Consider a border cycle C of the lowerleft region containing the central node (see the blue cycle in Figure 3(a)). Cycle C has 2D nodes. For D � 2k + 1, k ∈ N. Let Q be the horizontal line passing through the edge bc and R be the vertical line passing through the node c of the triangle T in the graph L D that divide the graph L D into four regions (upper left, upper right, lower left, and lower right), as shown in Figure 3. Consider a border cycle C of the lower-left region containing the edge bc(see the blue cycle in Figure 3(b)). e cycle C has 2D nodes.

Values for Δ � 7
Theorem 3. Let D be an even natural number, and let G be an infinite enhanced mesh network. en, Proof. For D � 2, the graph contains K 1,7 , which is the biggest subgraph of G of valency 7, since in the graph L D (shown in Figure 3(a)) of maximal valency Δ � 8, the central node can be attached to at most 7 nodes. Hence, Figure 4) from L D as follows.
Let L be the central horizontal line passing through the central node n, M be the upper neighboring row, and N be the lower neighboring row. Let H be the wheel graph W 5 of L D lying on and above the central line at a distance zero on the right side of the central node n. Furthermore, suppose that x is the hub node of H:

Upper-left region
Lower-right region

Journal of Mathematics
Proof. For D � 1, the graph contains K 3 and this is a biggest induced subgraph of G of maximal valency 7. Hence, N G (7, 1) � 3.
For D � 2k + 1, k ∈ N, andΔ � 7, we construct a subgraph Q D (shown in Figure 5) from L D as follows.
Let L be the horizontal line passing through edge bc of the central triangle T of graph L D and M be the upper neighboring horizontal row. Let K be the wheel subgraph of L D containing the central triangle T: Proof. For D � 2, the graph contains K 1,6 . is is a biggest subgraph of G of maximal valency 6 since in the graph L D (shown in Figure 3(a)) of maximal valency Δ � 8, and the central node can be connected to at most 6 nodes. Hence, Figure 6) from L D as follows.
Let L and M be the horizontal and vertical lines passing through the central node n that divide L D into four regions (upper left, upper right, lower left, and lower right). Let H and K be the wheel graphs W 5 of L D lying in the upper-right and lower-left regions at a distance 0 from the central node n. Furthermore, suppose that x and y be the hub vertices of H and K, respectively: Proof. For D � 1, the graph contains K 3 and this is a biggest induced subgraph of G of maximal valency 6. Hence, N G (6, 1) � 3. For D � 3, the graph L D of maximal valency 8 is a closed ball of radius 3 (shown Figure 3(b)). For maximal valency Δ � 6, the biggest subgraph cannot contain all the nodes of the graph L D , otherwise D � 4. us, the graph shown in Figure 7(a) is the biggest subgraph of maximal valency Δ � 6. Hence, N G (6, 3) � 13.
For D � 2k + 1, k ∈ N, and Δ � 6, we construct a subgraph Q D (shown in Figure 7) from L D as follows.
Let S be the wheel graph W 5 that contains the central triangle T. Let R be the horizontal row of the wheel graphs W 5 containing the wheel graph S. Let S 1 (S 2 ) be the wheel graphs lying on left (right) of S and is adjacent to it. Also, let S 3 be the wheel graph W 5 below the wheel graph S and is adjacent to it: Proof. For D � 2, the graph contains K 1,5 . is is the biggest subgraph of G of maximal valency 5, since in graph L D (shown in Figure 3(a)) of maximal valency Δ � 8, the central node can be connected to at most 5 nodes. Hence, N G (5, 2) � 6.
For D � 4, the graph shown in Figure 8 is of maximal valency 5 and diameter 4, which is the subgraph of L D (shown in Figure 3 Proof. For D � 1, the graph contains K 3 and this is a biggest induced subgraph of G of maximal valency 5. Hence, For D � 3, the graph shown in Figure 9(a) is of maximal valency Δ � 5 and diameter 3, which is a subgraph of L D (shown in Figure 3(b)). Hence, N G (5, 3) ≥ 11.
For D � 5, the graph shown in Figure 9(a) is of maximal valency Δ � 5 and diameter 5, which is a subgraph of L D (shown in Figure 3(b)). Hence, N G (5, 3) ≥ 33.
Let L be the horizontal line passing through the edge bc of the central triangle T and R be the vertical column of wheel graphs W 5 containing the central triangle T that divide the graph L D into four regions (upper left, upper right, lower left, and lower right), as shown in Figure 9(b). Let K be the wheel graph W 5 that contains the central triangle T. Let K 1 and K 3 be the wheel graphs W 5 lying in the lower-left and lower-right regions, respectively, at a distance zero from the central triangle T, both lying on the line L. Let K 2 be the wheel graph W 5 lying below the wheel graph K and is adjacent to it:  e resulting subgraph is denoted by Q D (shown in Figure 9, for D � 7, 9), which is the spanning subgraph of L D ; hence, V(L D ) � V(Q D ). Furthermore, the graph Q D also has diameter D since the distance of all the nodes of Q D from the central triangle is the same as in L D except the nodes x, y, and z whose distance from central triangle T is ≤((D − 1)/2). is implies that Q D is the biggest subgraph of G of maximal valency Δ � 5 and diameter D. For even D, we prove the succeeding theorem.
Proof. e lower bound for N G (4, 2) is 5, since the graph contains K 1,4 . e biggest subgraph of G of maximal valency 8 is the graph L D (shown Figure 3(a)). When Δ � 4, the central node n of L D can be connected to at most four vertices. erefore, we have to remove at least four vertices, which are adjacent to n, otherwise D > 2. is implies that the biggest subgraph of G of maximal valency 4 can contain at most 5 vertices. Hence, N G (4, 2) � 5.
For D � 4, the graph G contains the subgraph (shown in Figure 10(a)) that has maximal valency and diameter 4. us, N G (4, D) ≥ 17. For D � 6, the graph G contain the subgraph (shown in Figure 10(a)) that has maximal valency 4 and diameter 6. lower-right regions, respectively, at a distance 1 from the central node n and lying on the line L. Let P 1 be the path of length D passing through the central node n and lies in upper-left and lower-right regions. Let P 2 be the path of length D passing through the central node n and lies in upper-right and lower-left regions. Furthermore, suppose that the nodes u and v lie on the path P 1 and w and x lie on the path P 2 that are adjacent to the central node n:

Journal of Mathematics
Proof. For D � 1, the graph contains K 3 and this is a biggest induced subgraph of G of maximal valency 4. Hence, N G (4, 1) � 3. For D � 3, the graph shown in Figure 11(a) is of maximal valency Δ � 4 and diameter 4, which is a subgraph of L D (shown in Figure 3(b)). Hence, N G (4, 3) ≥ 9.
For D � 5, the graph shown in Figure 11(a) is of maximal valency Δ � 4 and diameter 5, which is a subgraph of L D (shown in Figure 3(b)). Hence, N G (4, 5) ≥ 27.
For D � 2k + 1, k ∈ N − 1, 2, 3 { }, and Δ � 4, we construct a subgraph Q D (shown in Figure 11) from L D as follows Let L be the horizontal line passing through the edge bc of the central triangle T and R be the vertical column of wheel graphs W 5 containing the central triangle T that divide the graph L D into four regions (upper left, upper right, lower left, and lower right). Let K be the wheel graph W 5 that contains the central triangle T. Let K 1 and K 3 be the wheel graphs W 5 lying in the lower-left and lowerright regions, respectively, at a distance 1 from the central triangle T, both lying on the line L. Let K 2 be the wheel graph W 5 lying below the wheel graph K and is adjacent to it. Let K 4 (K 6 ) be the wheel graph W 5 lying on left (right) of the wheel graph K and is adjacent to it. Let K 5 be the wheel graph W 5 lying above the wheel graph K and is adjacent to it: hub edges in each W 5 lying in the column R and above K except K 5 (ix) Delete the lower-left, lower-right, and upper-left hub edges in each W 5 lying in the column R and below K except K 2 (x) Delete the upper-left, upper-right, and lower-right hub edges in K 2 (xi) Delete the lower-left, lower-right, and upper-left hub edges in K 5 (xii) Furthermore, delete the edges yb, wb, tc, and uc (xiii) Now, reinstate the lower horizontal edges in K 1 and K 3 (xiv) Now, reinstate the upper horizontal edges in K 4 and K 6 e resulting subgraph is denoted by Q D (shown in Figure 11(b), for D � 9), which is the spanning subgraph of L D ; hence, V(L D ) � V(Q D ). Furthermore, the graph Q D also has diameter D since the distance of all the nodes of Q D from the central triangle is the same as in L D except the nodes s, t, u, v, w, x, y, and z whose distance from central triangle T is ≤((D − 1)/2). is implies that Q D is the biggest subgraph of G of maximal valency Δ � 4 and diameter D. □ 4.6. Bounds for Δ � 3. For D � 1, the graph contains K 3 and this is a biggest induced subgraph of G of maximal valency 3. Hence, N G (3, 1) � 3. e lower bound for N G (3, 2) is 4, since the graph contains K 1,3 . e biggest subgraph of G of maximal valency 8 is the graph L D (shown in Figure 3(a)). When Δ � 3, the central node n of L D can be connected to at most three nodes. erefore, we have to remove at least three nodes which are adjacent to n , otherwise D > 2. is implies that the biggest subgraph of G of maximal valency 3 can contain at most 4 nodes. Hence, N G (3, 2) � 4.
Proof. For D � 4, 6, 8, the subgraphs of G shown in Figure 12 are of maximal valency Δ � 3. It is also easy to check that the diameter of each graph is D. Hence, the theorem is true for D � 4, 6, 8. For D � 2k + 8, L D is the biggest subgraph of G of maximal valency Δ � 8 and diameter D. Let W D be the subgraph of L D of maximal valency Δ � 3 and diameter D.
e structure of the graph W D for D � 10, 12, 14 is depicted in Figure 12. e central node of graph W D correspond to the central node of L D and is exhibited by •. e whole graph W D is constructed from L D by deleting those nodes which have distance greater than (D/2) from the central node n. e structure of the deleted nodes in the construction of W D is shown in Figure 12. To calculate the deleted nodes of L D , we draw horizontal lines on deleted nodes and counting them in horizontal rows from top to bottom: Delete vertices in L D � 2 + 3 + 3 + 3 + · · · { to(k + 1) terms} + 4 + (2k + 8) + 2 + 5 + 4 + 3 + 3 + 3 + · · · to(k + 1) { terms} + 2 � 8k + 33.
For D � 2k + 9, L D is the biggest subgraph of G of maximal valency Δ � 8 and diameter D. Let W D be the subgraph of L D of maximal valency Δ � 3 and diameter D.
e structure of the graph W D for D � 11, 13, 15 is exhibited in Figure 13. e central triangle of W D correspond to the central triangle T of L D and is depicted by •. e whole graph W D is constructed from L D by deleting those nodes which have distance greater than ((D − 1)/2) from the central triangle T. e structure of the deleted nodes in the construction of W D is shown in Figure 13. Now, calculate the deleted nodes of L D in horizontal rows from top to bottom.
Furthermore, d(x, T) ≤ ((D − 1)/2) for every node x ∈ W D . us, W D has diameter at most D.
Hence, the theorem is satisfied for D � 2k + 9, k ∈ N.
Using the result from eorems 11 and 12 and the fact that N G (Δ − 1, D) ≤ N G (Δ, D) for any graph, we get the succeeding statement. (17)

Conclusion
In the Degree Diameter Problem, the largest graphs in term of vertices are computed with given degree and diameter. In this work, we have considered the restricted version of Degree Diameter Problem which states that given connected undirected host graph G, an upper bound D for the maximum degree, and an upper bound Δ for the diameter find the largest connected subgraph of maximum degree ≤ Δ and diameter ≤ D. is problem becomes of particular interest when we consider the host graph as the network. e topology of a network (such as a telecommunications, multiprocessor, or local area network, to name just a  Figure 13: e subgraphs of G for D � 3, 5, 7, 9, 11, 13, 15 and Δ � 3. few) is usually modeled by a graph in which vertices represent 'nodes' (stations or processors) while undirected or directed edges stand for 'links' or other types of connections.
ere are many important features in the designing of such networks. One of the important aspects is to put limitation on vertex degree and its diameter. ese two parameters in networks are interpreted as follows. By the degree of a node, it is meant to have the number of connections attached to that node; on the contrary, the diameter shows the largest number of links that must be required to transmit a message between any two nodes. e natural question that arises in this case is "what is then the largest number of nodes in a network with a limited degree and diameter?" If we design the network so that there is no directed edge, then this leads to the Degree/Diameter Problem.
Planar graphs are popular for network design because they have a convenient physical layout. Of all planar network topologies, the (square) mesh is the most popular and this has been studied in [1,[3][4][5]7]. In this work, we have extended this study by considering the host graph G as the graph obtained from the enhanced mesh network.
In future, it will be interesting to find more such mesh networks and study the optimal subgraphs. Furthermore, we can extend this study to directed networks.

Data Availability
Data from previous studies were used to support this study.
ey are cited at relevant places within the text as references.

Conflicts of Interest
e authors declare that they have no conflicts of interest.