Solutions of Detour Distance Graph Equations

Graph theory is a useful mathematical structure used to model pairwise relations between sensor nodes in wireless sensor networks. Graph equations are nothing but equations in which the unknown factors are graphs. Many problems and results in graph theory can be formulated in terms of graph equations. In this paper, we solved some graph equations of detour two-distance graphs, detour three-distance graphs, detour antipodal graphs involving with the line graphs.


Introduction
The recent rapid growth in the Internet of things has necessitated the development of new approaches to persistent issues in wireless sensor networks. These issues include minimum obstacles in the end-to-end communication path, location accuracy, latency, and delay, among others. These problems can be mitigated by using the distance graph to create local algorithms, i.e., algorithms with minimum communication rounds. We study distance graph applications in wireless sensor networks with a focus on minimum path obstacles and high localization accuracy.
A wireless sensor network (WSN) is a network of tiny wireless sensors that can sense a parameter of interest. The sensed data is forwarded to a base station through the formed ad hoc network of sensor nodes. There are many application areas of WSNs, including M2M communication and the Internet of Things (IoT). WSNs are a fundamental building block of smart homes, smart workplaces, and smart cities, among others. It has a lot of other essential purposes for modern technology, such as scientific research, rescue operations, and scientific discoveries. As sensors are a power constraint tiny devices, energy conservation for extending the network's lifetime is a challenging issue. A wireless sensor network's lifetime heavily depends on innovative schemes that mitigate energy consumption. The distance graph is used to form and localize an ad hoc network of sensor nodes so that sensed data can be forwarded to the base station with little energy cost.
Therefore, finding the solutions to these graph equations is essential. A lot of research has been done by many researchers during the past fifty years, and their results have made a significant contribution in graph theory.
Let X be a nontrivial finite connected graph. Every graph X with detour distance D defines a metric space. The n-distance graph of X, denoted by T n (X), is the graph with V(T n (X)) = V(X) in which two vertices u and v are adjacent in T n (X) if and only if d X (u, v) = n. Furthermore, for each set S ⊆ dist(V(X); D), the detour distance graph Γ D (V(X), S) is the graph having V(X) as its vertex set and two vertices u and v in this graph are adjacent if and only if D(u, v) ∈ S. We denote this graph simply by D (X, S). The graph D n (X) := D (X, {n}) is called the detour n-distance graph of X. If n equals the detour diameter of X, then this graph is called the detour antipodal graph of X, which is denoted by DA (G). A graph X is said to be a detour self n-distance graph if D n (X) ∼ = X.
The line graph L(G) of a graph G is the graph whose vertices correspond to the edges of G and wherein two vertices are adjacent in L(G) if and best if the corresponding edges are adjacent in G.
In this work, we consider the detour distance graphs. Specifically, we solve the graph equations involving detour two-distance graphs, detour three-distance graphs, detour antipodal distance graphs, line graphs, and the complement of graphs. In addition, we solve some equations involving two-distance graphs, three-distance graphs, and antipodal distance graphs with the graphs mentioned earlier.
Harary, Heode, and Kedlacek first studied the two-distance graph. They investigated the connectedness of two-distance graphs. This graph and the relationship between the two-distance graph and line graph was further studied in [1][2][3][4][5][6][7][8][9]. In 2014, Ali Azimi and Mohammad D. X solved the graph equations T 2 (X) ∼ = P n and T 2 (X) ∼ = C n . In 2015, Ramuel P. Ching and Garces gave three characterizations of two-distance graphs and found all the graphs X such that T 2 (X) ∼ = kP 2 or K m ∪ K n , which can be found in [10]. S. K Simic and some mathematicians solved some graph equations of line graphs, which can be found in [11][12][13][14][15][16][17]. In 2017, R. Rajkumar and S. Celine Prabha solved some graph equations of two-distance, three-distance and n-distance graph equations, which can be found in [18][19][20]. In 2018, R. Rajkumar and S. Celine Prabha found the characterization of the distance graph of a path which was described in [21].
Motivated by the results listed above, we solved some graph equations of distance graphs and line graphs. Other graph-theoretic terms and notations that are not explicitly defined here can be found in [2].

Solutions of Graph Equations of Detour Two-Distance Graphs and Line Graphs
First, we consider some graph equations of type D 2 (G) ∼ = G 1 ∪ G 2 , where G 1 and G 2 are given graphs and investigate the solution G of these equations.
The main result of solving graph equations involving detour two-distance graphs we prove in this section is the following: Theorem 1. Let G be a graph. Then, If G is connected, then Lemma 1. Let n ≥ 3 be an integer. Then, Let v be a vertex in G which is not in the cycle of G. Without loss of generality, we may assume that v is adjacent to v 1 . Then, If G is the graph other than these graphs, then by the argument of part (a), By the structure of G, the graph L(G) is connected non-unicyclic. Let the maximum length among all the cycles in L(G) be k. Clearly k ≥ 4. If k ≥ 5, then any vertex in such a cycle is isolated in D 2 (L(G)). If k = 4, then G is the graph obtained from K 3 by adding a pendent edge to any of its vertex. In this case, ) has two isolated vertices. However, G has exactly one isolated vertex. Thus, If G is the graph other than these graphs, then the maximum length among all the cycles in L(G) is at least six. Then, any vertex in such a cycle is isolated in D 2 (L(G)). Therefore, D 2 (L(G)) G.

6.
By part (e), D 2 (L(G)) is disconnected. Thus, by a similar argument as in part (d), we get D 2 (L(G)) G.
If G is the graph other than these graphs, then by the argument of part (a), D 2 (G) is disconnected and so T 2 (D 2 (L(G))) is disconnected. Hence, L(G) is connected. Therefore, T 2 (D 2 (L(G))) L(G).
Combining Propositions 1-3, we get the proof of Theorem 1.

Solutions of Graph Equations of Detour Three-Distance Graphs and Line Graphs
The main result on solving graph equations involving three-distance graphs we prove in this section is the following: Theorem 2. Let G be a graph. Then, If G is connected, then To prove the above theorem, we start with the following: Lemma 2. Let n ≥ 4 be an integer. Then, If n ≥ 7, then the detour distance between any two vertices of C n is at least four. Therefore, D 3 (C n ) ∼ = K n . Proposition 4. Let n ≥ 4 be an integer. Then, 1.
Proposition 5. Let G be a graph.

Proof. 1.
Let G be connected non-unicyclic. Suppose that D 3 (L(G)) ∼ = G. Then, we have |V(G)| = |E(G)|, so G is unicyclic, since G is connected, which is a contradiction to our assumption that G is non-unicyclic. Thus, D 3 (L(G)) G. The proof of the rest of the cases are similar to the above.

Proof. 1.
By the structure of G, the graph L(G) is connected non-unicyclic. Let the maximum length among all the cycles in L(G) be k. Clearly, k ≥ 4. If k ≥ 7, then any vertex in such a cycle is isolated in D 3 (L(G)). If k = 4, 5, 6, then D 3 (L(G)) ∼ = C 4 , C 5 and 3K 2 , respectively. Thus, D 3 (L(G)) is disconnected and D 3 (L(G)) G.

2.
By part (a), D 3 (L(G)) is disconnected. However, G is connected except for the graph C 3 (r, 0, , 0), r ≥ 1. Thus, D 3 (L(G)) G. If G ∼ = C 3 (r, 0, , 0), r ≥ 1, then D 3 (L(G)) has two isolated vertices. However, G has exactly one isolated vertex. Therefore, If G is the graph other than these graphs, then the maximum length among all the cycles in L(G) is at least seven. Then, any vertex in such a cycle is isolated in D 3 (L(G)). Thus, D 3 (L(G)) G.

4.
By part (c), D 3 (L(G)) is disconnected. Therefore, the rest of the proof is similar to part (b).
The proof is similar to the proof of part (b), since D 3 (D 3 (L(G))), T 2 (D 3 (L(G))) and T 3 (D 3 (L(G))) are disconnected. (i)-(j): The proof of part (i) and (j) are similar to the proof of parts (c) and (d), respectively, since D 3 (L(G)) is disconnected.
Combining Propositions 4-6, we get the proof of Theorem 2.

Solutions of Graph Equations of Detour Antipodal Graphs and Line Graphs
The main result on solving graph equations involving detour antipodal graphs we prove in this section is the following: T 2 (DA (L(G))) ∼ = G if and only if G ∼ = C n , where n ≥ 5 and n is odd; 10. T 2 (DA (G)) ∼ = G if and only if G ∼ = C n , where n ≥ 5 and n is odd;
if and only if n = 3, 5.
If G is the graph other than these graphs, then by the argument of part (a), DA (G) is disconnected. Thus, L(G) is connected. Hence, none of D 2 (DA (G)), D 3 (DA (G)), T 2 (DA (G)), T 3 (DA (G)) and A(DA (G)) is isomorphic to L(G).
Combining Propositions 7-9, we get the proof of Theorem 3.

Conclusions
Given a set of wireless sensor nodes and connections, graph theory provides a useful tool to simplify the many moving parts of dynamic systems. In this work, we mainly focused on the study of detour distance graph equations. In particular, we solved some graph equations involving detour two-distance graphs, detour three-distance graphs, detour antipodal graphs and line graphs. This solution is believed to be useful for many researchers and businesses working in wireless sensor networks.