On the Diameter and Incidence Energy of Iterated Total Graphs
Abstract
:1. Introduction
2. Diameter of Total Graphs
2.1. Main Results
- If are vertices of G, then we can assume that is a vertex of G, . Then, , which is impossible.
- If are vertices of the line graph of G, that is to say if are edges of G, then we can assume that is an edge of G, . Thus, , which is impossible.
- If is a vertex of G and is an edge of G, say , we can assume without loss of generality that is a vertex of or is an edge of . Suppose that is a vertex of . Then, . Since , then . In addition, . Thereby, is a diameter subgraph of G some , where , which is impossible. Otherwise, suppose is an edge of , with then . Since , then . Moreover, . Thus, is a diameter subgraph of G for some , where , which is impossible.That is, for any pair of vertices in , which is a contradiction.
- , , and are induced subgraphs of G, and
- is a diameter path of G, and
- is a diameter subgraph of G for some , where .
- Case 1: Suppose . If , then is a diameter path in G—a contradiction. If , then is an induced subgraph of G—a contradiction.
- Case 2: Suppose , then there must exist edges in G, say , such that is incident to in G, , and . Let , . Then
- (i)
- If is not adjacent to and is not adjacent to , then is an induced subgraph of G—a contradiction.
- (ii)
- If is adjacent to (or is adjacent to ), then is an induced subgraph of G—a contradiction.
- (iii)
- If is adjacent to and is adjacent to , then is an induced subgraph of G—a contradiction.
- Case 3: Suppose that is a diameter subgraph of G for some , where . If , then is a diameter subgraph of G for some , where —a contradiction. If , then is a diameter path of G—a contradiction. If , then is an induced subgraph of G—a contradiction.
- or or is an induced subgraph of G, or
- is a diameter path of G, or
- is a diameter subgraph of G for some , where .
- or , or is an induced subgraph of G, or
- is a diameter subgraph of G.
- (i)
- If is not adjacent to , is not adjacent to and is not adjacent to , then is an induced subgraph of G. Moreover, since , then is a diameter path of G for some v. Hence, is a diameter subgraph of G.
- (ii)
- If is adjacent to (or is adjacent to ) and is not adjacent to , then is an induced subgraph of G.
- (iii)
- If is adjacent to , is adjacent to and is not adjacent to , then is an induced subgraph of G.
- (iv)
- If is adjacent to , is not adjacent to and is not adjacent to , then is an induced subgraph of G.
- a.
- if and only if .
- b.
- If the following conditions fail to hold:
- , and is an induced subgraph of G, and
- is a diameter subgraph of G,
then if and only if or .
- a.
- If , then all the vertices of are adjacent. That is, . Thus, .Conversely, let and . Then, there exists two different edges in G, say and . Thus, . Therefore, —a contradiction.
- b.
- Suppose that none of the three graphs , , and are induced subgraphs of G, and that is not a diameter subgraph of G. By Lemma 3 it does not occur that . Now, if , by Lemma 1 some of the following cases are verified:
- Case 1: . Then, , thus any couple of edges of G are incidents. Consequently, or . Since , we concluded that .
- Case 2: . Then, , thus any couple of vertices of G are adjacents. Therefore, .
- Case 3: is a diameter subgraph of G. Then, . Thus, .
Conversely, let . Then, any couple of vertices in G are at distance 1. Let and be two different edges in G. If and are incident edges, then . Otherwise, since is an edge in G, we have . Finally, let be an edge of G and let v be a vertex of G. If or , then . Otherwise, since is an edge of G, then . Hence, .If , then all the edges of G are incidents to a common vertex. Therefore, all vertices are pairwise incident in and thus . Hence, . Moreover, all vertices are adjacents to a common vertex in G. Then, . Finally, since , then any edge of G and any vertex of G are to distance less than or equal to 2. Therefore, .
2.2. Results for Iterated Total Graphs
- or or is an induced subgraph of , or
- is a diameter path of , or
- is a diameter subgraph of for some , where .
- (a)
- If is an induced subgraph of , then either or is an induced subgraph of or is an induced subgraph of . Since , then is an induced subgraph of .
- (b)
- If is an induced subgraph of , then is an induced subgraph of . By , is an induced subgraph of .
- (c)
- If is an induced subgraph of , then is an induced subgraph of . By , is an induced subgraph of .
- (d)
- If is a diameter path of , then either or is a diameter path of or is a diameter path of . Since , then is an induced subgraph of .
- (e)
- If is a diameter subgraph of , for some , where . Then, is a diameter path of . By , is a diameter path of .
2.3. Results for Iterated Line Graphs
- a.
- If and such that
- b.
- If such that or or is an induced subgraph of G. Then
- a.
- Suppose . By Lemma 4, or or is an induced subgraph of . Then, is an induced subgraph of . Since . Then, is an induced subgraph of . Thus, is an induced subgraph of . Then, is an induced subgraph of . Following this process, we concluded that is an induced subgraph of G.
- b.
- Suppose or or is an induced subgraph of G. By Lemma 2, or or is an induced subgraph of . Moreover, , , and . By Lemma 4, .
3. Energy of Iterated Graphs
3.1. Incidence Energy of Iterated Graphs
- 1.
- , and
- 2.
- .
- Let v be a vertex of the k-th iterated total graph of G. Then
- Case a: If v is a vertex of the -th iterated total graph of G, then v is adjacent to vertices and incident to edges in . Thus, the degree of v in is
- Case b: If v is an edge of the -th iterated total graph of G, then v is adjacent in each extreme to edges and incident to its two extreme vertices in . Thus, the degree of v in is
Therefore, the k-th iterated total graph of G is a regular graph of degree - Let be a regular graph with edges, then . Therefore, the order of the k-th iterated total graph of G is
- and
- .
- , and
- .
- , and
- .
3.2. An Application: Constructing Non-isomorphic Cospectral Graphs
- (a)
- and are of the same order, and have the same edges number.
- (b)
- and are cospectral if and only if and are cospectral.
4. Conclusions
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
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Lenes, E.; Mallea-Zepeda, E.; Robbiano, M.; Rodríguez, J. On the Diameter and Incidence Energy of Iterated Total Graphs. Symmetry 2018, 10, 252. https://doi.org/10.3390/sym10070252
Lenes E, Mallea-Zepeda E, Robbiano M, Rodríguez J. On the Diameter and Incidence Energy of Iterated Total Graphs. Symmetry. 2018; 10(7):252. https://doi.org/10.3390/sym10070252
Chicago/Turabian StyleLenes, Eber, Exequiel Mallea-Zepeda, María Robbiano, and Jonnathan Rodríguez. 2018. "On the Diameter and Incidence Energy of Iterated Total Graphs" Symmetry 10, no. 7: 252. https://doi.org/10.3390/sym10070252
APA StyleLenes, E., Mallea-Zepeda, E., Robbiano, M., & Rodríguez, J. (2018). On the Diameter and Incidence Energy of Iterated Total Graphs. Symmetry, 10(7), 252. https://doi.org/10.3390/sym10070252