Signed Complete Graphs with Maximum Index
Let Γ = (G, σ) be a signed graph, where G is the underlying simple graph and σ E(G) → {−, +} is the sign function on the edges of G. The adjacency matrix of a signed graph has −1 or +1 for adjacent vertices, depending on the sign of the edges. It was conjectured that if is a signed complete graph of...
| Main Authors: | , , , |
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| Format: | Article |
| Language: | English |
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University of Zielona Góra
2020-05-01
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| Series: | Discussiones Mathematicae Graph Theory |
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| Online Access: | https://doi.org/10.7151/dmgt.2276 |
| _version_ | 1797757678322188288 |
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| author | Akbari Saieed Dalvandi Soudabeh Heydari Farideh Maghasedi Mohammad |
| author_facet | Akbari Saieed Dalvandi Soudabeh Heydari Farideh Maghasedi Mohammad |
| author_sort | Akbari Saieed |
| collection | DOAJ |
| description | Let Γ = (G, σ) be a signed graph, where G is the underlying simple graph and σ E(G) → {−, +} is the sign function on the edges of G. The adjacency matrix of a signed graph has −1 or +1 for adjacent vertices, depending on the sign of the edges. It was conjectured that if is a signed complete graph of order n with k negative edges, k ---lt--- n − 1 and has maximum index, then negative edges form K1,k. In this paper, we prove this conjecture if we confine ourselves to all signed complete graphs of order n whose negative edges form a tree of order k + 1. A [1, 2]-subgraph of G is a graph whose components are paths and cycles. Let Γ be a signed complete graph whose negative edges form a [1, 2]-subgraph. We show that the eigenvalues of Γ satisfy the following inequalities: |
| first_indexed | 2024-03-12T18:19:04Z |
| format | Article |
| id | doaj.art-3f734e71e7e8492dbdcd3984b2ae4efc |
| institution | Directory Open Access Journal |
| issn | 2083-5892 |
| language | English |
| last_indexed | 2024-03-12T18:19:04Z |
| publishDate | 2020-05-01 |
| publisher | University of Zielona Góra |
| record_format | Article |
| series | Discussiones Mathematicae Graph Theory |
| spelling | doaj.art-3f734e71e7e8492dbdcd3984b2ae4efc2023-08-02T08:59:13ZengUniversity of Zielona GóraDiscussiones Mathematicae Graph Theory2083-58922020-05-0140239340310.7151/dmgt.2276dmgt.2276Signed Complete Graphs with Maximum IndexAkbari Saieed0Dalvandi Soudabeh1Heydari Farideh2Maghasedi Mohammad3Department of Mathematical Science, Sharif University of Technology, Tehran, IranDepartment of Mathematics Karaj Branch, Islamic Azad University, Karaj, IranDepartment of Mathematics Karaj Branch, Islamic Azad University, Karaj, IranDepartment of Mathematics Karaj Branch, Islamic Azad University, Karaj, IranLet Γ = (G, σ) be a signed graph, where G is the underlying simple graph and σ E(G) → {−, +} is the sign function on the edges of G. The adjacency matrix of a signed graph has −1 or +1 for adjacent vertices, depending on the sign of the edges. It was conjectured that if is a signed complete graph of order n with k negative edges, k ---lt--- n − 1 and has maximum index, then negative edges form K1,k. In this paper, we prove this conjecture if we confine ourselves to all signed complete graphs of order n whose negative edges form a tree of order k + 1. A [1, 2]-subgraph of G is a graph whose components are paths and cycles. Let Γ be a signed complete graph whose negative edges form a [1, 2]-subgraph. We show that the eigenvalues of Γ satisfy the following inequalities:https://doi.org/10.7151/dmgt.2276signed graphcomplete graphindex05c2205c50 |
| spellingShingle | Akbari Saieed Dalvandi Soudabeh Heydari Farideh Maghasedi Mohammad Signed Complete Graphs with Maximum Index Discussiones Mathematicae Graph Theory signed graph complete graph index 05c22 05c50 |
| title | Signed Complete Graphs with Maximum Index |
| title_full | Signed Complete Graphs with Maximum Index |
| title_fullStr | Signed Complete Graphs with Maximum Index |
| title_full_unstemmed | Signed Complete Graphs with Maximum Index |
| title_short | Signed Complete Graphs with Maximum Index |
| title_sort | signed complete graphs with maximum index |
| topic | signed graph complete graph index 05c22 05c50 |
| url | https://doi.org/10.7151/dmgt.2276 |
| work_keys_str_mv | AT akbarisaieed signedcompletegraphswithmaximumindex AT dalvandisoudabeh signedcompletegraphswithmaximumindex AT heydarifarideh signedcompletegraphswithmaximumindex AT maghasedimohammad signedcompletegraphswithmaximumindex |