Date of Publication

2025

Document Type

Bachelor's Thesis

Degree Name

Bachelor of Science in Mathematics with Specialization in Computer Applications

Subject Categories

Mathematics

College

College of Science

Department/Unit

Mathematics and Statistics Department

Thesis Advisor

Daryl Q. Granario

Defense Panel Chair

John Vincent S. Morales

Defense Panel Member

April Lynne D. Say-awen

Abstract (English)

This paper investigates vertex sedentariness in path, cycle, and cactus graphs through the lens of continuous-time quantum walks and spectral decomposition. A vertex is said to be sedentary when a quantum state that begins at that vertex tends to remain there over time. Cactus graphs, defined as connected graphs whose cycles intersect at most at a single vertex, vary in complexity depending on their number of vertices. Our study explores how unitary evolution and spectral properties determine which vertices exhibit sedentariness, and how these behaviors are preserved under direct products of unitary operators. We first establish parity-based behavior in cycle graphs: even cycles are non-sedentary, odd cycles are sedentary, and C6 behaves as a special periodic but non-sedentary case. Path graphs are shown to be computationally challenging due to their non-periodic and irrational spectra. Extending these methods to cactus graphs up to six vertices, we analyze their adjacency matrices and spectral data, offering insights and future directions for studying sedentariness in broader graph families.

Abstract Format

html

Abstract (Filipino)

Ang papel na ito ay nag-iimbestiga ng vertex sedentariness sa path, cycle, at cactus na grap gamit ang continuous-time quantum walks at spectral decomposition. Ang isang vertex ay itinuturing na sedentary kapag ang quantum state na nagsimula dito ay may probabilidad na manatili sa paglipas ng panahon. Ang cactus graphs, na binubuo ng mga konektadong grap na ang mga cycles ay nagtatagpo lamang sa isang vertex, ay nag-iiba sa antas ng komplikasyon depende sa bilang ng vertices. Sinusuri ng pag-aaral kung paano ito kumikilos pag lumipas ang panahon at mga spectral properties na nagtatakda kung aling vertices ang nagiging sedentary. Itinatag namin ang parity-based behavior sa cycle na grap: ang even cycles ay non-sedentary, ang odd cycles ay sedentary, at ang C6 ay isang espesyal na kasong periodic ngunit non-sedentary. Ipinapakita rin na ang path na grap ay mahirap suriin dahil sa kanilang non-periodic at irrational spectra nito. Pinalawak namin ang mga metodong ito sa cactus graphs hanggang anim na vertices upang suriin ang kanilang adjacency matrices at spectral data, nagbibigay ng mahahalagang pananaw para sa mas malawak na pag-aaral ng sedentariness sa iba’t ibang pamilya ng grap.

Abstract Format

html

Language

English

Format

Electronic

Keywords

Graph theory

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Embargo Period

12-14-2025

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