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5.1: Discovering Graphs

( \newcommand{\kernel}{\mathrm{null}\,}\)

Definitions are typically constructed after we work with new objects. The definition is constructed to match the properties we have observed and need. In this section we will practice this by developing a definition for a type of object known as a graph.

isgraph1.svgisgraph5.svg
isgraph2.svgisgraph6.svg
isgraph3.svgisgraph7.svg
isgraph4.svgisgraph8.svg
Figure 5.1.1: Each of these is a graph
notgraph1.svgnotgraph5.svg
notgraph2.svgnotgraph6.svg
notgraph3.svgnotgraph7.svg
notgraph4.svg
Figure 5.1.2: None of these is a graph

Checkpoint 5.1.3

Based on the examples in Figure 5.1.1 and Figure 5.1.2 determine which of the following are graphs.

isitgraph1.svgisitgraph5.svg
isitgraph2.svgisitgraph6.svg
isitgraph3.svgisitgraph7.svg
isitgraph4.svgisitgraph8.svg

Example 5.1.4: Each of these is a graph.

Each of these is a graph.

  1. P={p1,p2,p3}Q={q1,q2,q3}q1={p1,p2}q2={p1,p3}q3={p2,p3}
  2. P={p1,p2,p3,p4}Q=
  3. P={p1,p2,p3,p4}Q={q1,q2}q1={p1,p3}q2={p2,p4}
  4. P={p1,p2,p3,p4,p5}L={1,2,3,4,5,6}1={p2,p3}2={p1,p4}3={p4,p5}4={p1,p5}5={p2,p4}6={p2,p5}
  5. V={c,d,e,f}E={{c,d},{d,e},{e,f},{c,f}}
  6. V={v1,v2,v3,v4}E={{v1,v2},{v1,v3},{v2,v3},{v2,v4}}
  7. P={p}Q=
  8. P={q,r,s,t}Q={{q,r},{r,t},{q,t},{r,s}}

Example 5.1.5

None of these is a graph.

  1. P={p1,p2,p3}Q={q1,q2}q1={p1,p2}q2={p3,p4}
  2. P={p1,p2}Q={q1,q2}q1={p1,p2}q2={p1,p2}
  3. P={p1,p2,p3}Q={q1,q2,q3}q1&=(p1,p2)q2=(p1,p3)q3=(p2,p3)
  4. V={x,y,z}E={f,g,h}f=(x,y)g=(y,x)h=(y,z)
  5. P=Q=
  6. P&={q,r,s,t}Q={{q,r},{r,t},{s}}
  7. V={p1,p2,,pn,}E={{p1,p2},{p2,p4},{p1,p3},{p4,p6}}
  8. P={p1,p2,p3}Q={q1,q2,q3}q1={p1,p2}q2={p1,p3}

Checkpoint 5.1.6

Based on the examples in Example 5.1.4 and Example 5.1.5 determine which of the following are graphs.

  1. P={p1,p2,p3}Q={q1}q1={p1,p3}
  2. P={p1,p2,p3}Q={q1,q2}q1={p1,p2}
  3. V={v1,v2,,vn,}E={e1,e2,,en,}e1=(v1,v2)en=(vn,vn+1)
  4. P={p,q,r}Q={t,u,v}t={p,q}u={p,r}v={q,r}
  5. P={p1,p2,p3}Q=
  6. V={u1,u2,u3,u4}E={(u1,u2),(u2,u1),(u3,u4)}
  7. V={v1,v2,v3}E={{v1,v2},{v1,v3},{v3}}
  8. P={p,q,r,s}Q={{p,q},{p,r},{p,s},{q,r},{q,s},{r,s}}
Checkpoint 5.1.7
In terms of the diagrams in Figure 5.1.1 list properties required and properties not allowed in graphs.
Checkpoint 5.1.8
In terms of the sets in Example 5.1.4 list properties required and properties not allowed in graphs.
Checkpoint 5.1.9
Draw the diagram form of the first graph in Example 5.1.4.
Checkpoint 5.1.10
Write the set form of the fourth graph (bottom left) in Figure 5.1.1.

This page titled 5.1: Discovering Graphs is shared under a GNU Free Documentation License 1.3 license and was authored, remixed, and/or curated by Mark A. Fitch via source content that was edited to the style and standards of the LibreTexts platform.

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