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Sonst wäre die Gradsumme ungerade.",[565,766,767],{},"Der Name kommt von der Deutung als Party: Knoten sind Gäste, Kanten Händedrücke. 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Die Summe ist ",[578,864,866,892],{"className":865},[581],[578,867,869],{"className":868},[585],[587,870,871],{"xmlns":589},[591,872,873,889],{},[594,874,875,878,881,883,886],{},[710,876,877],{},"12",[601,879,880],{},"=",[710,882,712],{},[601,884,885],{},"⋅",[710,887,888],{},"6",[615,890,891],{"encoding":617},"12 = 2 \\cdot 6",[578,893,895,915,935],{"className":894,"ariaHidden":623},[622],[578,896,898,901,904,908,912],{"className":897},[627],[578,899],{"className":900,"style":724},[631],[578,902,877],{"className":903},[649],[578,905],{"className":906,"style":907},[830],"margin-right:0.2778em;",[578,909,880],{"className":910},[911],"mrel",[578,913],{"className":914,"style":907},[830],[578,916,918,921,924,928,932],{"className":917},[627],[578,919],{"className":920,"style":724},[631],[578,922,712],{"className":923},[649],[578,925],{"className":926,"style":927},[830],"margin-right:0.2222em;",[578,929,885],{"className":930},[931],"mbin",[578,933],{"className":934,"style":927},[830],[578,936,938,941],{"className":937},[627],[578,939],{"className":940,"style":724},[631],[578,942,888],{"className":943},[649],", also hat er ",[578,946,948,961],{"className":947},[581],[578,949,951],{"className":950},[585],[587,952,953],{"xmlns":589},[591,954,955,959],{},[594,956,957],{},[710,958,888],{},[615,960,888],{"encoding":617},[578,962,964],{"className":963,"ariaHidden":623},[622],[578,965,967,970],{"className":966},[627],[578,968],{"className":969,"style":724},[631],[578,971,888],{"className":972},[649]," Kanten.",[565,975,976,979,980,1009,1010,1038,1039,1113],{},[569,977,978],{},"Unmöglicher Graph:"," Kann jeder von ",[578,981,983,997],{"className":982},[581],[578,984,986],{"className":985},[585],[587,987,988],{"xmlns":589},[591,989,990,995],{},[594,991,992],{},[710,993,994],{},"5",[615,996,994],{"encoding":617},[578,998,1000],{"className":999,"ariaHidden":623},[622],[578,1001,1003,1006],{"className":1002},[627],[578,1004],{"className":1005,"style":724},[631],[578,1007,994],{"className":1008},[649]," Gästen genau ",[578,1011,1013,1026],{"className":1012},[581],[578,1014,1016],{"className":1015},[585],[587,1017,1018],{"xmlns":589},[591,1019,1020,1024],{},[594,1021,1022],{},[710,1023,789],{},[615,1025,789],{"encoding":617},[578,1027,1029],{"className":1028,"ariaHidden":623},[622],[578,1030,1032,1035],{"className":1031},[627],[578,1033],{"className":1034,"style":724},[631],[578,1036,789],{"className":1037},[649]," anderen die Hand geben? Die Gradsumme wäre ",[578,1040,1042,1065],{"className":1041},[581],[578,1043,1045],{"className":1044},[585],[587,1046,1047],{"xmlns":589},[591,1048,1049,1062],{},[594,1050,1051,1053,1055,1057,1059],{},[710,1052,994],{},[601,1054,885],{},[710,1056,789],{},[601,1058,880],{},[710,1060,1061],{},"15",[615,1063,1064],{"encoding":617},"5 \\cdot 3 = 15",[578,1066,1068,1086,1104],{"className":1067,"ariaHidden":623},[622],[578,1069,1071,1074,1077,1080,1083],{"className":1070},[627],[578,1072],{"className":1073,"style":724},[631],[578,1075,994],{"className":1076},[649],[578,1078],{"className":1079,"style":927},[830],[578,1081,885],{"className":1082},[931],[578,1084],{"className":1085,"style":927},[830],[578,1087,1089,1092,1095,1098,1101],{"className":1088},[627],[578,1090],{"className":1091,"style":724},[631],[578,1093,789],{"className":1094},[649],[578,1096],{"className":1097,"style":907},[830],[578,1099,880],{"className":1100},[911],[578,1102],{"className":1103,"style":907},[830],[578,1105,1107,1110],{"className":1106},[627],[578,1108],{"className":1109,"style":724},[631],[578,1111,1061],{"className":1112},[649],", also ungerade. Das ist unmöglich.",[565,1115,1116,1211,1212,1240,1241,1295],{},[569,1117,1118,1119,1210],{},"Vollständiger Graph ",[578,1120,1122,1143],{"className":1121},[581],[578,1123,1125],{"className":1124},[585],[587,1126,1127],{"xmlns":589},[591,1128,1129,1140],{},[594,1130,1131],{},[1132,1133,1134,1137],"msub",{},[597,1135,1136],{},"K",[597,1138,1139],{},"n",[615,1141,1142],{"encoding":617},"K_n",[578,1144,1146],{"className":1145,"ariaHidden":623},[622],[578,1147,1149,1153],{"className":1148},[627],[578,1150],{"className":1151,"style":1152},[631],"height:0.8333em;vertical-align:-0.15em;",[578,1154,1156,1160],{"className":1155},[649],[578,1157,1136],{"className":1158,"style":1159},[649,650],"margin-right:0.0715em;",[578,1161,1164],{"className":1162},[1163],"msupsub",[578,1165,1169,1201],{"className":1166},[1167,1168],"vlist-t","vlist-t2",[578,1170,1173,1196],{"className":1171},[1172],"vlist-r",[578,1174,1178],{"className":1175,"style":1177},[1176],"vlist","height:0.1514em;",[578,1179,1181,1186],{"style":1180},"top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;",[578,1182],{"className":1183,"style":1185},[1184],"pstrut","height:2.7em;",[578,1187,1193],{"className":1188},[1189,1190,1191,1192],"sizing","reset-size6","size3","mtight",[578,1194,1139],{"className":1195},[649,650,1192],[578,1197,1200],{"className":1198},[1199],"vlist-s","​",[578,1202,1204],{"className":1203},[1172],[578,1205,1208],{"className":1206,"style":1207},[1176],"height:0.15em;",[578,1209],{},":"," Jeder der ",[578,1213,1215,1228],{"className":1214},[581],[578,1216,1218],{"className":1217},[585],[587,1219,1220],{"xmlns":589},[591,1221,1222,1226],{},[594,1223,1224],{},[597,1225,1139],{},[615,1227,1139],{"encoding":617},[578,1229,1231],{"className":1230,"ariaHidden":623},[622],[578,1232,1234,1237],{"className":1233},[627],[578,1235],{"className":1236,"style":692},[631],[578,1238,1139],{"className":1239},[649,650]," Knoten ist mit allen anderen verbunden, hat also Grad ",[578,1242,1244,1264],{"className":1243},[581],[578,1245,1247],{"className":1246},[585],[587,1248,1249],{"xmlns":589},[591,1250,1251,1261],{},[594,1252,1253,1255,1258],{},[597,1254,1139],{},[601,1256,1257],{},"−",[710,1259,1260],{},"1",[615,1262,1263],{"encoding":617},"n - 1",[578,1265,1267,1286],{"className":1266,"ariaHidden":623},[622],[578,1268,1270,1274,1277,1280,1283],{"className":1269},[627],[578,1271],{"className":1272,"style":1273},[631],"height:0.6667em;vertical-align:-0.0833em;",[578,1275,1139],{"className":1276},[649,650],[578,1278],{"className":1279,"style":927},[830],[578,1281,1257],{"className":1282},[931],[578,1284],{"className":1285,"style":927},[830],[578,1287,1289,1292],{"className":1288},[627],[578,1290],{"className":1291,"style":724},[631],[578,1293,1260],{"className":1294},[649],". Daraus folgt",[578,1297,1300],{"className":1298},[1299],"katex-display",[578,1301,1303,1361],{"className":1302},[581],[578,1304,1306],{"className":1305},[585],[587,1307,1309],{"xmlns":589,"display":1308},"block",[591,1310,1311,1358],{},[594,1312,1313,1317,1320,1322,1324,1343,1345],{},[597,1314,1316],{"mathvariant":1315},"normal","∣",[597,1318,1319],{},"E",[597,1321,1316],{"mathvariant":1315},[601,1323,880],{},[1325,1326,1327,1341],"mfrac",{},[594,1328,1329,1331,1333,1335,1337,1339],{},[597,1330,1139],{},[601,1332,607],{"stretchy":606},[597,1334,1139],{},[601,1336,1257],{},[710,1338,1260],{},[601,1340,613],{"stretchy":606},[710,1342,712],{},[601,1344,880],{},[594,1346,1347,1349,1356],{},[601,1348,607],{"fence":623},[1325,1350,1352,1354],{"linethickness":1351},"0px",[597,1353,1139],{},[710,1355,712],{},[601,1357,613],{"fence":623},[615,1359,1360],{"encoding":617},"|E| = \\frac{n(n-1)}{2} = \\binom{n}{2}",[578,1362,1364,1389,1497],{"className":1363,"ariaHidden":623},[622],[578,1365,1367,1370,1373,1377,1380,1383,1386],{"className":1366},[627],[578,1368],{"className":1369,"style":632},[631],[578,1371,1316],{"className":1372},[649],[578,1374,1319],{"className":1375,"style":1376},[649,650],"margin-right:0.0576em;",[578,1378,1316],{"className":1379},[649],[578,1381],{"className":1382,"style":907},[830],[578,1384,880],{"className":1385},[911],[578,1387],{"className":1388,"style":907},[830],[578,1390,1392,1396,1488,1491,1494],{"className":1391},[627],[578,1393],{"className":1394,"style":1395},[631],"height:2.113em;vertical-align:-0.686em;",[578,1397,1399,1403,1485],{"className":1398},[649],[578,1400],{"className":1401},[645,1402],"nulldelimiter",[578,1404,1406],{"className":1405},[1325],[578,1407,1409,1476],{"className":1408},[1167,1168],[578,1410,1412,1473],{"className":1411},[1172],[578,1413,1416,1429,1440],{"className":1414,"style":1415},[1176],"height:1.427em;",[578,1417,1419,1423],{"style":1418},"top:-2.314em;",[578,1420],{"className":1421,"style":1422},[1184],"height:3em;",[578,1424,1426],{"className":1425},[649],[578,1427,712],{"className":1428},[649],[578,1430,1432,1435],{"style":1431},"top:-3.23em;",[578,1433],{"className":1434,"style":1422},[1184],[578,1436],{"className":1437,"style":1439},[1438],"frac-line","border-bottom-width:0.04em;",[578,1441,1443,1446],{"style":1442},"top:-3.677em;",[578,1444],{"className":1445,"style":1422},[1184],[578,1447,1449,1452,1455,1458,1461,1464,1467,1470],{"className":1448},[649],[578,1450,1139],{"className":1451},[649,650],[578,1453,607],{"className":1454},[645],[578,1456,1139],{"className":1457},[649,650],[578,1459],{"className":1460,"style":927},[830],[578,1462,1257],{"className":1463},[931],[578,1465],{"className":1466,"style":927},[830],[578,1468,1260],{"className":1469},[649],[578,1471,613],{"className":1472},[655],[578,1474,1200],{"className":1475},[1199],[578,1477,1479],{"className":1478},[1172],[578,1480,1483],{"className":1481,"style":1482},[1176],"height:0.686em;",[578,1484],{},[578,1486],{"className":1487},[655,1402],[578,1489],{"className":1490,"style":907},[830],[578,1492,880],{"className":1493},[911],[578,1495],{"className":1496,"style":907},[830],[578,1498,1500,1504],{"className":1499},[627],[578,1501],{"className":1502,"style":1503},[631],"height:2.4em;vertical-align:-0.95em;",[578,1505,1507,1516,1562],{"className":1506},[649],[578,1508,1512],{"className":1509,"style":1511},[645,1510],"delimcenter","top:0em;",[578,1513,607],{"className":1514},[1515,1191],"delimsizing",[578,1517,1519],{"className":1518},[1325],[578,1520,1522,1554],{"className":1521},[1167,1168],[578,1523,1525,1551],{"className":1524},[1172],[578,1526,1529,1540],{"className":1527,"style":1528},[1176],"height:1.1076em;",[578,1530,1531,1534],{"style":1418},[578,1532],{"className":1533,"style":1422},[1184],[578,1535,1537],{"className":1536},[649],[578,1538,712],{"className":1539},[649],[578,1541,1542,1545],{"style":1442},[578,1543],{"className":1544,"style":1422},[1184],[578,1546,1548],{"className":1547},[649],[578,1549,1139],{"className":1550},[649,650],[578,1552,1200],{"className":1553},[1199],[578,1555,1557],{"className":1556},[1172],[578,1558,1560],{"className":1559,"style":1482},[1176],[578,1561],{},[578,1563,1565],{"className":1564,"style":1511},[655,1510],[578,1566,613],{"className":1567},[1515,1191],[565,1569,1570,1571,1599,1600,1737],{},"Bei ",[578,1572,1574,1587],{"className":1573},[581],[578,1575,1577],{"className":1576},[585],[587,1578,1579],{"xmlns":589},[591,1580,1581,1585],{},[594,1582,1583],{},[710,1584,888],{},[615,1586,888],{"encoding":617},[578,1588,1590],{"className":1589,"ariaHidden":623},[622],[578,1591,1593,1596],{"className":1592},[627],[578,1594],{"className":1595,"style":724},[631],[578,1597,888],{"className":1598},[649]," Gästen, die sich alle die Hand geben, sind das ",[578,1601,1603,1631],{"className":1602},[581],[578,1604,1606],{"className":1605},[585],[587,1607,1608],{"xmlns":589},[591,1609,1610,1628],{},[594,1611,1612,1624,1626],{},[1325,1613,1614,1622],{},[594,1615,1616,1618,1620],{},[710,1617,888],{},[601,1619,885],{},[710,1621,994],{},[710,1623,712],{},[601,1625,880],{},[710,1627,1061],{},[615,1629,1630],{"encoding":617},"\\frac{6 \\cdot 5}{2} = 15",[578,1632,1634,1728],{"className":1633,"ariaHidden":623},[622],[578,1635,1637,1641,1719,1722,1725],{"className":1636},[627],[578,1638],{"className":1639,"style":1640},[631],"height:1.1901em;vertical-align:-0.345em;",[578,1642,1644,1647,1716],{"className":1643},[649],[578,1645],{"className":1646},[645,1402],[578,1648,1650],{"className":1649},[1325],[578,1651,1653,1707],{"className":1652},[1167,1168],[578,1654,1656,1704],{"className":1655},[1172],[578,1657,1660,1675,1683],{"className":1658,"style":1659},[1176],"height:0.8451em;",[578,1661,1663,1666],{"style":1662},"top:-2.655em;",[578,1664],{"className":1665,"style":1422},[1184],[578,1667,1669],{"className":1668},[1189,1190,1191,1192],[578,1670,1672],{"className":1671},[649,1192],[578,1673,712],{"className":1674},[649,1192],[578,1676,1677,1680],{"style":1431},[578,1678],{"className":1679,"style":1422},[1184],[578,1681],{"className":1682,"style":1439},[1438],[578,1684,1686,1689],{"style":1685},"top:-3.394em;",[578,1687],{"className":1688,"style":1422},[1184],[578,1690,1692],{"className":1691},[1189,1190,1191,1192],[578,1693,1695,1698,1701],{"className":1694},[649,1192],[578,1696,888],{"className":1697},[649,1192],[578,1699,885],{"className":1700},[931,1192],[578,1702,994],{"className":1703},[649,1192],[578,1705,1200],{"className":1706},[1199],[578,1708,1710],{"className":1709},[1172],[578,1711,1714],{"className":1712,"style":1713},[1176],"height:0.345em;",[578,1715],{},[578,1717],{"className":1718},[655,1402],[578,1720],{"className":1721,"style":907},[830],[578,1723,880],{"className":1724},[911],[578,1726],{"className":1727,"style":907},[830],[578,1729,1731,1734],{"className":1730},[627],[578,1732],{"className":1733,"style":724},[631],[578,1735,1061],{"className":1736},[649]," Händedrücke.",[565,1739,1740,576,1743,1772,1773,1801,1802,973],{},[569,1741,1742],{},"Würfel als Graph:",[578,1744,1746,1760],{"className":1745},[581],[578,1747,1749],{"className":1748},[585],[587,1750,1751],{"xmlns":589},[591,1752,1753,1758],{},[594,1754,1755],{},[710,1756,1757],{},"8",[615,1759,1757],{"encoding":617},[578,1761,1763],{"className":1762,"ariaHidden":623},[622],[578,1764,1766,1769],{"className":1765},[627],[578,1767],{"className":1768,"style":724},[631],[578,1770,1757],{"className":1771},[649]," Ecken mit je Grad ",[578,1774,1776,1789],{"className":1775},[581],[578,1777,1779],{"className":1778},[585],[587,1780,1781],{"xmlns":589},[591,1782,1783,1787],{},[594,1784,1785],{},[710,1786,789],{},[615,1788,789],{"encoding":617},[578,1790,1792],{"className":1791,"ariaHidden":623},[622],[578,1793,1795,1798],{"className":1794},[627],[578,1796],{"className":1797,"style":724},[631],[578,1799,789],{"className":1800},[649]," ergeben ",[578,1803,1805,1833],{"className":1804},[581],[578,1806,1808],{"className":1807},[585],[587,1809,1810],{"xmlns":589},[591,1811,1812,1830],{},[594,1813,1814,1826,1828],{},[1325,1815,1816,1824],{},[594,1817,1818,1820,1822],{},[710,1819,1757],{},[601,1821,885],{},[710,1823,789],{},[710,1825,712],{},[601,1827,880],{},[710,1829,877],{},[615,1831,1832],{"encoding":617},"\\frac{8 \\cdot 3}{2} = 12",[578,1834,1836,1925],{"className":1835,"ariaHidden":623},[622],[578,1837,1839,1842,1916,1919,1922],{"className":1838},[627],[578,1840],{"className":1841,"style":1640},[631],[578,1843,1845,1848,1913],{"className":1844},[649],[578,1846],{"className":1847},[645,1402],[578,1849,1851],{"className":1850},[1325],[578,1852,1854,1905],{"className":1853},[1167,1168],[578,1855,1857,1902],{"className":1856},[1172],[578,1858,1860,1874,1882],{"className":1859,"style":1659},[1176],[578,1861,1862,1865],{"style":1662},[578,1863],{"className":1864,"style":1422},[1184],[578,1866,1868],{"className":1867},[1189,1190,1191,1192],[578,1869,1871],{"className":1870},[649,1192],[578,1872,712],{"className":1873},[649,1192],[578,1875,1876,1879],{"style":1431},[578,1877],{"className":1878,"style":1422},[1184],[578,1880],{"className":1881,"style":1439},[1438],[578,1883,1884,1887],{"style":1685},[578,1885],{"className":1886,"style":1422},[1184],[578,1888,1890],{"className":1889},[1189,1190,1191,1192],[578,1891,1893,1896,1899],{"className":1892},[649,1192],[578,1894,1757],{"className":1895},[649,1192],[578,1897,885],{"className":1898},[931,1192],[578,1900,789],{"className":1901},[649,1192],[578,1903,1200],{"className":1904},[1199],[578,1906,1908],{"className":1907},[1172],[578,1909,1911],{"className":1910,"style":1713},[1176],[578,1912],{},[578,1914],{"className":1915},[655,1402],[578,1917],{"className":1918,"style":907},[830],[578,1920,880],{"className":1921},[911],[578,1923],{"className":1924,"style":907},[830],[578,1926,1928,1931],{"className":1927},[627],[578,1929],{"className":1930,"style":724},[631],[578,1932,877],{"className":1933},[649],[560,1935,1937],{"id":1936},"typische-fehler","Typische Fehler",[1939,1940,1941,2078,2142],"ul",{},[1942,1943,1944,1947,1948,2019,2020,2049,2050,973],"li",{},[569,1945,1946],{},"Den Faktor 2 vergessen."," Die Gradsumme ist nicht die Kantenzahl. Bei ",[578,1949,1951,1969],{"className":1950},[581],[578,1952,1954],{"className":1953},[585],[587,1955,1956],{"xmlns":589},[591,1957,1958,1966],{},[594,1959,1960],{},[1132,1961,1962,1964],{},[597,1963,1136],{},[710,1965,888],{},[615,1967,1968],{"encoding":617},"K_6",[578,1970,1972],{"className":1971,"ariaHidden":623},[622],[578,1973,1975,1978],{"className":1974},[627],[578,1976],{"className":1977,"style":1152},[631],[578,1979,1981,1984],{"className":1980},[649],[578,1982,1136],{"className":1983,"style":1159},[649,650],[578,1985,1987],{"className":1986},[1163],[578,1988,1990,2011],{"className":1989},[1167,1168],[578,1991,1993,2008],{"className":1992},[1172],[578,1994,1997],{"className":1995,"style":1996},[1176],"height:0.3011em;",[578,1998,1999,2002],{"style":1180},[578,2000],{"className":2001,"style":1185},[1184],[578,2003,2005],{"className":2004},[1189,1190,1191,1192],[578,2006,888],{"className":2007},[649,1192],[578,2009,1200],{"className":2010},[1199],[578,2012,2014],{"className":2013},[1172],[578,2015,2017],{"className":2016,"style":1207},[1176],[578,2018],{}," ist die Gradsumme ",[578,2021,2023,2037],{"className":2022},[581],[578,2024,2026],{"className":2025},[585],[587,2027,2028],{"xmlns":589},[591,2029,2030,2035],{},[594,2031,2032],{},[710,2033,2034],{},"30",[615,2036,2034],{"encoding":617},[578,2038,2040],{"className":2039,"ariaHidden":623},[622],[578,2041,2043,2046],{"className":2042},[627],[578,2044],{"className":2045,"style":724},[631],[578,2047,2034],{"className":2048},[649],", es gibt aber ",[578,2051,2053,2066],{"className":2052},[581],[578,2054,2056],{"className":2055},[585],[587,2057,2058],{"xmlns":589},[591,2059,2060,2064],{},[594,2061,2062],{},[710,2063,1061],{},[615,2065,1061],{"encoding":617},[578,2067,2069],{"className":2068,"ariaHidden":623},[622],[578,2070,2072,2075],{"className":2071},[627],[578,2073],{"className":2074,"style":724},[631],[578,2076,1061],{"className":2077},[649],[1942,2079,2080,2083,2084,2112,2113,2141],{},[569,2081,2082],{},"Schleifen einfach zählen."," Eine Schleife erhöht den Grad um ",[578,2085,2087,2100],{"className":2086},[581],[578,2088,2090],{"className":2089},[585],[587,2091,2092],{"xmlns":589},[591,2093,2094,2098],{},[594,2095,2096],{},[710,2097,712],{},[615,2099,712],{"encoding":617},[578,2101,2103],{"className":2102,"ariaHidden":623},[622],[578,2104,2106,2109],{"className":2105},[627],[578,2107],{"className":2108,"style":724},[631],[578,2110,712],{"className":2111},[649],", nicht um ",[578,2114,2116,2129],{"className":2115},[581],[578,2117,2119],{"className":2118},[585],[587,2120,2121],{"xmlns":589},[591,2122,2123,2127],{},[594,2124,2125],{},[710,2126,1260],{},[615,2128,1260],{"encoding":617},[578,2130,2132],{"className":2131,"ariaHidden":623},[622],[578,2133,2135,2138],{"className":2134},[627],[578,2136],{"className":2137,"style":724},[631],[578,2139,1260],{"className":2140},[649],".",[1942,2143,2144,2147,2148,2141],{},[569,2145,2146],{},"Auf gerichtete Graphen übertragen."," Dort gilt stattdessen: Summe der Eingangsgrade gleich Summe der Ausgangsgrade gleich ",[578,2149,2151,2169],{"className":2150},[581],[578,2152,2154],{"className":2153},[585],[587,2155,2156],{"xmlns":589},[591,2157,2158,2166],{},[594,2159,2160,2162,2164],{},[597,2161,1316],{"mathvariant":1315},[597,2163,1319],{},[597,2165,1316],{"mathvariant":1315},[615,2167,2168],{"encoding":617},"|E|",[578,2170,2172],{"className":2171,"ariaHidden":623},[622],[578,2173,2175,2178,2181,2184],{"className":2174},[627],[578,2176],{"className":2177,"style":632},[631],[578,2179,1316],{"className":2180},[649],[578,2182,1319],{"className":2183,"style":1376},[649,650],[578,2185,1316],{"className":2186},[649],{"title":2188,"searchDepth":2189,"depth":2189,"links":2190},"",2,[2191,2192,2193],{"id":562,"depth":2189,"text":563},{"id":770,"depth":2189,"text":771},{"id":1936,"depth":2189,"text":1937},[2195,2198,2200,2205,2207,2209,2211],{"lhs":2196,"rhs":2197},"3 + 2 + 2 + 3 + 2","2*6",{"lhs":2199,"rhs":1061},"5*3",{"lhs":2201,"rhs":2202,"where":2203},"n*(n - 1)","2*combinations(n, 2)",{"n":2204},6,{"lhs":2206,"rhs":1061},"6*5\u002F2",{"lhs":2208,"rhs":1061},"combinations(6, 2)",{"lhs":2210,"rhs":877},"8*3\u002F2",{"lhs":2212,"rhs":2213},"6*5","2*15","G = (V, E) \\text{ endlicher ungerichteter Graph; Schleifen zählen beim Grad doppelt}","\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>G\u003C\u002Fmi>\u003Cmo>=\u003C\u002Fmo>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmi>V\u003C\u002Fmi>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003Cmi>E\u003C\u002Fmi>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">G = (V, E)\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.6833em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">G\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">=\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.2222em;\">V\u003C\u002Fspan>\u003Cspan class=\"mpunct\">,\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0576em;\">E\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan> \u003Cspan class=\"katex\">endlicher ungerichteter Graph; Schleifen zählen beim Grad doppelt\u003C\u002Fspan>",null,"md","handshake-lemma","\\sum_{v \\in V} \\deg(v) = 2\\,|E|","\u003Cspan class=\"katex-display\">\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\" display=\"block\">\u003Csemantics>\u003Cmrow>\u003Cmunder>\u003Cmo>∑\u003C\u002Fmo>\u003Cmrow>\u003Cmi>v\u003C\u002Fmi>\u003Cmo>∈\u003C\u002Fmo>\u003Cmi>V\u003C\u002Fmi>\u003C\u002Fmrow>\u003C\u002Fmunder>\u003Cmi>deg\u003C\u002Fmi>\u003Cmo>⁡\u003C\u002Fmo>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmi>v\u003C\u002Fmi>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003Cmo>=\u003C\u002Fmo>\u003Cmn>2\u003C\u002Fmn>\u003Cmtext> \u003C\u002Fmtext>\u003Cmi mathvariant=\"normal\">∣\u003C\u002Fmi>\u003Cmi>E\u003C\u002Fmi>\u003Cmi mathvariant=\"normal\">∣\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">\\sum_{v \\in V} \\deg(v) = 2\\,|E|\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:2.3717em;vertical-align:-1.3217em;\">\u003C\u002Fspan>\u003Cspan class=\"mop op-limits\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:1.05em;\">\u003Cspan style=\"top:-1.8557em;margin-left:0em;\">\u003Cspan class=\"pstrut\" style=\"height:3.05em;\">\u003C\u002Fspan>\u003Cspan class=\"katex-sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\" style=\"margin-right:0.0359em;\">v\u003C\u002Fspan>\u003Cspan class=\"mrel mtight\">∈\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal mtight\" style=\"margin-right:0.2222em;\">V\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-3.05em;\">\u003Cspan class=\"pstrut\" style=\"height:3.05em;\">\u003C\u002Fspan>\u003Cspan>\u003Cspan class=\"mop op-symbol large-op\">∑\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:1.3217em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mop\">de\u003Cspan style=\"margin-right:0.0139em;\">g\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0359em;\">v\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">=\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">2\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">∣\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0576em;\">E\u003C\u002Fspan>\u003Cspan class=\"mord\">∣\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>","university",{},true,"\u002Fde\u002Fformulas\u002Fmath\u002Fdiscrete-math\u002Fhandschlaglemma",[],[2227,2228,2229],"euler-polyhedron","binomial-coefficient","pigeonhole-principle",{"title":548,"description":2188},"de\u002Fformulas\u002Fmath\u002Fdiscrete-math\u002Fhandschlaglemma","In jedem endlichen ungerichteten Graphen ist die Summe aller Knotengrade doppelt so groß wie die Anzahl der Kanten.",[2234,2235,2236,2237],"graphentheorie","knotengrad","kanten","graph",[2239,2243,2246],{"symbol":2240,"meaning":2241,"symbolHtml":2242},"G = (V, E)","Graph mit Knotenmenge V und Kantenmenge E","\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>G\u003C\u002Fmi>\u003Cmo>=\u003C\u002Fmo>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmi>V\u003C\u002Fmi>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003Cmi>E\u003C\u002Fmi>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">G = (V, E)\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.6833em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">G\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">=\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.2222em;\">V\u003C\u002Fspan>\u003Cspan class=\"mpunct\">,\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0576em;\">E\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",{"symbol":618,"meaning":2244,"symbolHtml":2245},"Grad des Knotens v, die Anzahl der Kanten, die an v enden","\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>deg\u003C\u002Fmi>\u003Cmo>⁡\u003C\u002Fmo>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmi>v\u003C\u002Fmi>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">\\deg(v)\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mop\">de\u003Cspan style=\"margin-right:0.0139em;\">g\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0359em;\">v\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",{"symbol":2168,"meaning":2247,"symbolHtml":2248},"Anzahl der Kanten","\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi mathvariant=\"normal\">∣\u003C\u002Fmi>\u003Cmi>E\u003C\u002Fmi>\u003Cmi mathvariant=\"normal\">∣\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">|E|\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">∣\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0576em;\">E\u003C\u002Fspan>\u003Cspan class=\"mord\">∣\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>","C491Deif3OJlTHgwh3tcj7qiP2Iz1BYhKixWIPR23Og",[2251,2254,2257],{"stem":2252,"title":2253,"category":187},"de\u002Fformulas\u002Fmath\u002Fdiscrete-math\u002Feulerscher-polyedersatz","Eulerscher Polyedersatz",{"stem":2255,"title":2256,"category":187},"de\u002Fformulas\u002Fmath\u002Fdiscrete-math\u002Fschubfachprinzip","Schubfachprinzip",{"stem":2258,"title":2259,"category":441},"de\u002Fformulas\u002Fmath\u002Fstochastics\u002Fbinomialkoeffizient","Binomialkoeffizient",1793824276450]