Distinguishing Number and Distinguishing Index of the Join of Two Graphs
The distinguishing number (index) D(G) (D'(G)) of a graph G is the least integer d such that G has an vertex labeling (edge labeling) with d labels that is preserved only by a trivial automorphism. In this paper we study the distinguishing number and the distinguishing index of the join of two...
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| Format: | Article |
| Language: | English |
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University of Kashan
2019-12-01
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| Series: | Mathematics Interdisciplinary Research |
| Subjects: | |
| Online Access: | https://mir.kashanu.ac.ir/article_102109_ff92223a27f0fd1dfbcd2f75fc2bc091.pdf |
| _version_ | 1797630833332322304 |
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| author | Saeid Alikhani Samaneh Soltani |
| author_facet | Saeid Alikhani Samaneh Soltani |
| author_sort | Saeid Alikhani |
| collection | DOAJ |
| description | The distinguishing number (index) D(G) (D'(G)) of a graph G is the least integer d such that G has an vertex labeling (edge labeling) with d labels that is preserved only by a trivial automorphism. In this paper we study the distinguishing number and the distinguishing index of the join of two graphs G and H, i.e., G+H. We prove that 0≤ D(G+H)-max{D(G),D(H)}≤ z, where z depends on the number of some induced subgraphs generated by some suitable partitions of V(G) and V(H). Let Gk be the k-th power of G with respect to the join product. We prove that if G is a connected graph of order n ≥ 2, then Gk has the distinguishing index 2, except D'(K2+K2)=3. |
| first_indexed | 2024-03-11T11:13:27Z |
| format | Article |
| id | doaj.art-66886be0fde64fad9c0d8f33486f5511 |
| institution | Directory Open Access Journal |
| issn | 2476-4965 |
| language | English |
| last_indexed | 2024-03-11T11:13:27Z |
| publishDate | 2019-12-01 |
| publisher | University of Kashan |
| record_format | Article |
| series | Mathematics Interdisciplinary Research |
| spelling | doaj.art-66886be0fde64fad9c0d8f33486f55112023-11-11T08:11:49ZengUniversity of KashanMathematics Interdisciplinary Research2476-49652019-12-014223925110.22052/mir.2020.133523.1102102109Distinguishing Number and Distinguishing Index of the Join of Two GraphsSaeid Alikhani0Samaneh Soltani1Department of Mathematics, Yazd University, Yazd, IranDepartment of Mathematics, Yazd University, Yazd, IranThe distinguishing number (index) D(G) (D'(G)) of a graph G is the least integer d such that G has an vertex labeling (edge labeling) with d labels that is preserved only by a trivial automorphism. In this paper we study the distinguishing number and the distinguishing index of the join of two graphs G and H, i.e., G+H. We prove that 0≤ D(G+H)-max{D(G),D(H)}≤ z, where z depends on the number of some induced subgraphs generated by some suitable partitions of V(G) and V(H). Let Gk be the k-th power of G with respect to the join product. We prove that if G is a connected graph of order n ≥ 2, then Gk has the distinguishing index 2, except D'(K2+K2)=3.https://mir.kashanu.ac.ir/article_102109_ff92223a27f0fd1dfbcd2f75fc2bc091.pdfdistinguishing indexdistinguishing numberjoin |
| spellingShingle | Saeid Alikhani Samaneh Soltani Distinguishing Number and Distinguishing Index of the Join of Two Graphs Mathematics Interdisciplinary Research distinguishing index distinguishing number join |
| title | Distinguishing Number and Distinguishing Index of the Join of Two Graphs |
| title_full | Distinguishing Number and Distinguishing Index of the Join of Two Graphs |
| title_fullStr | Distinguishing Number and Distinguishing Index of the Join of Two Graphs |
| title_full_unstemmed | Distinguishing Number and Distinguishing Index of the Join of Two Graphs |
| title_short | Distinguishing Number and Distinguishing Index of the Join of Two Graphs |
| title_sort | distinguishing number and distinguishing index of the join of two graphs |
| topic | distinguishing index distinguishing number join |
| url | https://mir.kashanu.ac.ir/article_102109_ff92223a27f0fd1dfbcd2f75fc2bc091.pdf |
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