Eigenvalue monotonicity of q-Laplacians of trees along a poset
Let T be a tree on n vertices with q-Laplacian matrix LTq. Let GTSn be the generalized tree shift poset on the set of unlabelled trees with n vertices. We prove that for all q∈R, going up on GTSn has the following effect: the spectral radius of LTq increases while the smallest eigenvalue of LTq decr...
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| Veröffentlicht in: | Linear algebra and its applications 2019-06, Vol.571, p.110-131 |
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| container_title | Linear algebra and its applications |
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| creator | Nagar, Mukesh Kumar |
| description | Let T be a tree on n vertices with q-Laplacian matrix LTq. Let GTSn be the generalized tree shift poset on the set of unlabelled trees with n vertices. We prove that for all q∈R, going up on GTSn has the following effect: the spectral radius of LTq increases while the smallest eigenvalue of LTq decreases. We also prove that for all q∈R with |q|≥1, going up on GTSn increases the second smallest eigenvalue of LTq. These generalize known results for eigenvalues of the Laplacian matrix of trees. As a corollary, we obtain consequences about the eigenvalues of q,t-Laplacians and exponential distance matrices of trees. |
| doi_str_mv | 10.1016/j.laa.2019.02.018 |
| format | Article |
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Let GTSn be the generalized tree shift poset on the set of unlabelled trees with n vertices. We prove that for all q∈R, going up on GTSn has the following effect: the spectral radius of LTq increases while the smallest eigenvalue of LTq decreases. We also prove that for all q∈R with |q|≥1, going up on GTSn increases the second smallest eigenvalue of LTq. These generalize known results for eigenvalues of the Laplacian matrix of trees. As a corollary, we obtain consequences about the eigenvalues of q,t-Laplacians and exponential distance matrices of trees.</description><identifier>ISSN: 0024-3795</identifier><identifier>EISSN: 1873-1856</identifier><identifier>DOI: 10.1016/j.laa.2019.02.018</identifier><language>eng</language><publisher>Amsterdam: Elsevier Inc</publisher><subject>[formula omitted] poset ; Apexes ; Eigenvalues ; Exponential distance matrix ; Interlace ; Linear algebra ; q-Laplacian ; Tree ; Trees</subject><ispartof>Linear algebra and its applications, 2019-06, Vol.571, p.110-131</ispartof><rights>2019 Elsevier Inc.</rights><rights>Copyright American Elsevier Company, Inc. 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We prove that for all q∈R, going up on GTSn has the following effect: the spectral radius of LTq increases while the smallest eigenvalue of LTq decreases. We also prove that for all q∈R with |q|≥1, going up on GTSn increases the second smallest eigenvalue of LTq. These generalize known results for eigenvalues of the Laplacian matrix of trees. 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Let GTSn be the generalized tree shift poset on the set of unlabelled trees with n vertices. We prove that for all q∈R, going up on GTSn has the following effect: the spectral radius of LTq increases while the smallest eigenvalue of LTq decreases. We also prove that for all q∈R with |q|≥1, going up on GTSn increases the second smallest eigenvalue of LTq. These generalize known results for eigenvalues of the Laplacian matrix of trees. As a corollary, we obtain consequences about the eigenvalues of q,t-Laplacians and exponential distance matrices of trees.</abstract><cop>Amsterdam</cop><pub>Elsevier Inc</pub><doi>10.1016/j.laa.2019.02.018</doi><tpages>22</tpages><oa>free_for_read</oa></addata></record> |
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| source | Elsevier ScienceDirect Journals; Elektronische Zeitschriftenbibliothek - Frei zugängliche E-Journals |
| subjects | [formula omitted] poset Apexes Eigenvalues Exponential distance matrix Interlace Linear algebra q-Laplacian Tree Trees |
| title | Eigenvalue monotonicity of q-Laplacians of trees along a poset |
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