PlaniMatin perspectmsen walks

dcircuits

Back to minimal spanningtrees

Start connectivity Menger's theorem

Maitre representten

Vertrees numbered b in c indexingentrees invectors

makes

associated to aGmpk graph G get a hatrx

TG LTG i j0 if ii j are

notadjacent

2 if id j areadjacent

to TE

Proposition Tok jofwalks aflength k

from

i to jI 2 3

I O l e Or

fi o il

3 0 If O

I no ii ii 3

Observatory

tr TI closed walks of length 2

2 edges Edgar9degree

formula

tr CTE s cloud walks of length 3all come from D s

f G triangles in G

cyclesat length 3subgraphs

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Minimalspannybeest

I

7

PnnisAlgonthyBank outfrom a gonerwhee

Start at a nlex v chosen arbitrarily

add u to our subgraph 14whichone are buildy

Eachstep add an edgeand incident

vortex of minimal

weight suchthat exactlyone ofits notices

Ceuds is inHalready

canhe until all nemangedgeshaebothends in H

H built so its alwaysconnected

maximal acyclic treeA

minimal weight fif not minimal

choose a minimal onewhoshares

first k edges wl H e e k inorder ofconsLotsof HD

Wl k maximal

H te kn has a cycleshe ti maximal

acyclic

cycle not on HE kn E H so Feo in cycle in H

natin H

in constdun eaRi i e k is stillacyclic.GreenH

w lo 3 w eatacectic

now H't en Eo is at least as light as It

and has one more edge in common

lutotomeettopio Conakily dots

Two compety conceptsofconnectivity

a robotness1fragility how many edgesneedto

be removed todisconnect

graph

redundancy howmanydistinctways can yougo

behey

denfhewtmh.es

trroomswe

DEI k G min l ofedgeswhoseremovalmakes G disconnected or Inuit

k G mill of nerfires whose removal

makes G disconnectedCartmel

Det today Gis Imd if it has a art vertex

theorem K G E K G E S G Scot minimaldynee ofa notex