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Največja večkratnost lastnih vrednosti grafa in njegove prisilne ničle : delo diplomskega seminarja
Dolenc, Peter (Author), Oblak, Polona (Mentor) More about this mentor... This link opens in a new window

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Abstract
V delu enostavnim grafom na končni množici vozlišč priredimo množico simetričnih realnih matrik, ki imajo neničeln element na mestu $(i,j)$ natanko tedaj, ko $i \ne j$ in v grafu $G$ obstaja povezava med vozliščema $i$ in $j$. Največjo večkratnost lastnih vrednosti grafa definiramo kot največjo možno večkratnost lastnih vrednosti iz pripadajoče množice matrik. Ta parameter označimo z $M(G)$. V delu definiramo tudi parameter $Z(G)$ iz grafa $G$ in pokažemo, da za vsak enostaven graf $G$ velja $M(G) \leq Z(G)$. Podrobneje študiramo grafe s prereznimi vozlišči in si ogledamo obnašanje parametrov $M(G)$ in $Z(G)$ za takšne grafe.

Language:Slovenian
Keywords:simetrične matrike, večkratnost lastne vrednosti, prisilne ničle grafa, širitev ranga, ničelna širitev, induciran podgraf, prerezno vozlišče
Work type:Final seminar paper (mb14)
Tipology:2.11 - Undergraduate Thesis
Organization:FMF - Faculty of Mathematics and Physics
Year:2018
UDC:519.17
COBISS.SI-ID:18478937 This link opens in a new window
Views:465
Downloads:252
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Secondary language

Language:English
Title:Maximum multiplicity of eigenvalues of a graph and it's zero forcing sets
Abstract:
For any simple graph with finite set of verticies we assign a set of real symmetric matrices, whose $(i,j)$th entry is non-zero whenever $i \ne j$ and $\{i,j\}$ is an edge in $G$. We define maximum multiplicity of eigenvalues of a graph to be the largest possible multiplicity of eigenvalues of matrices in that set. We denote this parameter by $M(G)$. We also define parameter $Z(G)$ and show that for any simple graph $G$, $M(G)\leq Z(G)$. We take a closer look at graphs with cut-vertices and study parameters $M(G)$ and $Z(G)$ for these graphs.

Keywords:symmetric matrices, multiplicity of an eigenvalue, zero forcing set, rank spread, zero spread, induced subgraph, cut-vertex

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