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PreprintarXiv licencearXiv

Connected graphs with minimum adjacency spectral gap

  • Lele Liu
  • Michael Tait
  • Yi Wang
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Abstract

Let G be a connected graph, and let λ1(G) > λ2(G) denote its two largest adjacency eigenvalues. The spectral gap of G is defined as the difference λ1(G) − λ2(G). For integers r≥ 2 and s≥ 0, the double kite DK(r,s) is formed by taking two vertex-disjoint copies of the complete graph Kr and joining one specified vertex of each clique to a path with s internal vertices. Stanić (2013) conjectured that every connected n-vertex graph with minimum adjacency spectral gap is a double kite. In this paper, we confirm this conjecture for sufficiently large n.

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Cite

BibTeXRIS
BibTeX
@misc{liu2026connected,
  title = {{Connected graphs with minimum adjacency spectral gap}},
  author = {Lele Liu and Michael Tait and Yi Wang},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.37774},
  eprint = {2609.37774},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.37774},
}
RIS
TY  - UNPB
TI  - Connected graphs with minimum adjacency spectral gap
AU  - Lele Liu
AU  - Michael Tait
AU  - Yi Wang
PY  - 2026
DA  - 2026/09/29
PB  - arXiv
DO  - 10.48550/arxiv.2609.37774
UR  - https://arxiv.org/abs/2609.37774
AB  - Let G be a connected graph, and let λ_1(G) > λ_2(G) denote its two largest adjacency eigenvalues. The spectral gap of G is defined as the difference λ_1(G) − λ_2(G). For integers r≥ 2 and s≥ 0, the double kite DK(r,s) is formed by taking two vertex-disjoint copies of the complete graph K_r and joining one specified vertex of each clique to a path with s internal vertices. Stanić (2013) conjectured that every connected n-vertex graph with minimum adjacency spectral gap is a double kite. In this paper, we confirm this conjecture for sufficiently large n.
ER  -