Spectral deviations of graphs
For the graphs GG and HH, the spectral deviation of HH from GG is defined as ϱG(H)=∑μ∈Hminλ∈G∣λ−μ∣,{\varrho }_{G}\left(H)=\sum _{\mu \in H}\mathop{\min }\limits_{\lambda \in G}| \lambda -\mu | , where ∈\in designates that the given number is an eigenvalue of the adjacency matrix of the correspondin...
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| Format: | Article |
| Language: | English |
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De Gruyter
2025-02-01
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| Series: | Special Matrices |
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| Online Access: | https://doi.org/10.1515/spma-2024-0030 |
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| author | Stanić Zoran |
| author_facet | Stanić Zoran |
| author_sort | Stanić Zoran |
| collection | DOAJ |
| description | For the graphs GG and HH, the spectral deviation of HH from GG is defined as ϱG(H)=∑μ∈Hminλ∈G∣λ−μ∣,{\varrho }_{G}\left(H)=\sum _{\mu \in H}\mathop{\min }\limits_{\lambda \in G}| \lambda -\mu | , where ∈\in designates that the given number is an eigenvalue of the adjacency matrix of the corresponding graph. In this study, we consider the problem of existence of a proper induced subgraph HH of a prescribed graph GG such that ϱG(H)=0{\varrho }_{G}\left(H)=0, and the problem of determination of all such subgraphs. We investigate these problems in the framework of Smith graphs and their induced subgraphs, graphs with small second largest eigenvalue, graphs with small number of either positive or distinct eigenvalues, integral graphs, and chain graphs. Our results can be interesting in the context of graphs with a fixed number of distinct eigenvalues, eigenvalue distribution, or spectral distances of graphs. |
| format | Article |
| id | doaj-art-e5f74817572e43be83648a3aa756773c |
| institution | Kabale University |
| issn | 2300-7451 |
| language | English |
| publishDate | 2025-02-01 |
| publisher | De Gruyter |
| record_format | Article |
| series | Special Matrices |
| spelling | doaj-art-e5f74817572e43be83648a3aa756773c2025-02-10T13:25:12ZengDe GruyterSpecial Matrices2300-74512025-02-01131426510.1515/spma-2024-0030Spectral deviations of graphsStanić Zoran0Faculty of Mathematics, University of Belgrade, Studentski trg 16, 11 000 Belgrade, SerbiaFor the graphs GG and HH, the spectral deviation of HH from GG is defined as ϱG(H)=∑μ∈Hminλ∈G∣λ−μ∣,{\varrho }_{G}\left(H)=\sum _{\mu \in H}\mathop{\min }\limits_{\lambda \in G}| \lambda -\mu | , where ∈\in designates that the given number is an eigenvalue of the adjacency matrix of the corresponding graph. In this study, we consider the problem of existence of a proper induced subgraph HH of a prescribed graph GG such that ϱG(H)=0{\varrho }_{G}\left(H)=0, and the problem of determination of all such subgraphs. We investigate these problems in the framework of Smith graphs and their induced subgraphs, graphs with small second largest eigenvalue, graphs with small number of either positive or distinct eigenvalues, integral graphs, and chain graphs. Our results can be interesting in the context of graphs with a fixed number of distinct eigenvalues, eigenvalue distribution, or spectral distances of graphs.https://doi.org/10.1515/spma-2024-0030bounded eigenvaluesdistinct eigenvaluesintegral graphchain graph05c50 |
| spellingShingle | Stanić Zoran Spectral deviations of graphs Special Matrices bounded eigenvalues distinct eigenvalues integral graph chain graph 05c50 |
| title | Spectral deviations of graphs |
| title_full | Spectral deviations of graphs |
| title_fullStr | Spectral deviations of graphs |
| title_full_unstemmed | Spectral deviations of graphs |
| title_short | Spectral deviations of graphs |
| title_sort | spectral deviations of graphs |
| topic | bounded eigenvalues distinct eigenvalues integral graph chain graph 05c50 |
| url | https://doi.org/10.1515/spma-2024-0030 |
| work_keys_str_mv | AT staniczoran spectraldeviationsofgraphs |