Lower bounds for the algebraic connectivity of graphs with specified subgraphs
The second smallest eigenvalue of the Laplacian matrix of a graph G is called the algebraic connectivity and denoted by a(G). We prove thata(G)>π2/3(p(12g(n1, n2, ..., np)2 − π2)/4g(n1, n2, ..., np)4 + 4(q − p)(3g(np + 1, np + 2, ..., nq)2 − π2)/g(np + 1, np + 2, ..., nq)4),holds for every non-tr...
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| Format: | EJournal Article |
| Published: |
GTA Research Group, Univ. Newcastle, Indonesian Combinatorics Society and ITB,
2021-10-16.
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| Online Access: | Get Fulltext |
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| Summary: | The second smallest eigenvalue of the Laplacian matrix of a graph G is called the algebraic connectivity and denoted by a(G). We prove thata(G)>π2/3(p(12g(n1, n2, ..., np)2 − π2)/4g(n1, n2, ..., np)4 + 4(q − p)(3g(np + 1, np + 2, ..., nq)2 − π2)/g(np + 1, np + 2, ..., nq)4),holds for every non-trivial graph G which contains edge-disjoint spanning subgraphs G1, G2, ..., Gq such that, for 1 ≤ i ≤ p, a(Gi)≥a(Pni), with ni ≥ 2, and, for p + 1 ≤ i ≤ q, a(Gi)≥a(Cni), where Pni and Cni denote the path and the cycle of the corresponding order, respectively, and g denotes the geometric mean of given arguments. Among certain consequences, we emphasize the following lower bounda(G)>π212(4q − 3p)n2 − (16q − 15p)π2/12n4,referring to G which has n (n ≥ 2) vertices and contains p Hamiltonian paths and q − p Hamiltonian cycles, such that all of them are edge-disjoint. We also discuss the quality of the obtained lower bounds. |
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| Item Description: | https://www.ejgta.org/index.php/ejgta/article/view/995 |