{"id":2890,"date":"2025-06-07T03:11:45","date_gmt":"2025-06-07T03:11:45","guid":{"rendered":"https:\/\/diznr.com\/?p=2890"},"modified":"2025-06-07T03:11:45","modified_gmt":"2025-06-07T03:11:45","slug":"discrete-mathematics-for-gate-relation-b-w-vertices-and-edges-in-graph-simple","status":"publish","type":"post","link":"https:\/\/www.reilsolar.com\/pdf\/discrete-mathematics-for-gate-relation-b-w-vertices-and-edges-in-graph-simple\/","title":{"rendered":"Discrete mathematics for gate- Relation b\/w Vertices and Edges in Simple graph"},"content":{"rendered":"<p>Discrete mathematics for gate- Relation b\/w Vertices and Edges in Simple graph<\/p>\n<p>[fvplayer id=&#8221;160&#8243;]<\/p>\n<h3 data-start=\"0\" data-end=\"99\"><strong data-start=\"4\" data-end=\"97\">Discrete Mathematics for GATE \u2013 Relationship Between Vertices and Edges in a Simple Graph<\/strong><\/h3>\n<p data-start=\"101\" data-end=\"356\">In <strong data-start=\"104\" data-end=\"120\">graph theory<\/strong>, a <strong data-start=\"124\" data-end=\"140\">simple graph<\/strong> is an unweighted, undirected graph that <strong data-start=\"181\" data-end=\"230\">does not contain multiple edges or self-loops<\/strong>. The relationship between <strong data-start=\"257\" data-end=\"287\">vertices (V) and edges (E)<\/strong> in a simple graph follows several important properties and theorems.<\/p>\n<h3 data-start=\"363\" data-end=\"391\"><strong data-start=\"367\" data-end=\"391\">1. Basic Terminology<\/strong><\/h3>\n<ul data-start=\"392\" data-end=\"971\">\n<li data-start=\"392\" data-end=\"558\">\n<p data-start=\"394\" data-end=\"439\"><strong data-start=\"394\" data-end=\"408\">Graph (G):<\/strong> A pair <span class=\"katex\"><span class=\"katex-mathml\">G=(V,E)G = (V, E)<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">G<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">V<\/span><span class=\"mpunct\">,<\/span><span class=\"mord mathnormal\">E<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span> where:<\/p>\n<ul data-start=\"442\" data-end=\"558\">\n<li data-start=\"442\" data-end=\"492\"><span class=\"katex\"><span class=\"katex-mathml\">VV<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">V<\/span><\/span><\/span><\/span> is a finite set of <strong data-start=\"471\" data-end=\"491\">vertices (nodes)<\/strong>.<\/li>\n<li data-start=\"495\" data-end=\"558\"><span class=\"katex\"><span class=\"katex-mathml\">EE<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">E<\/span><\/span><\/span><\/span> is a set of <strong data-start=\"517\" data-end=\"557\">edges (connections between vertices)<\/strong>.<\/li>\n<\/ul>\n<\/li>\n<li data-start=\"559\" data-end=\"636\">\n<p data-start=\"561\" data-end=\"636\"><strong data-start=\"561\" data-end=\"578\">Simple Graph:<\/strong> A graph with <strong data-start=\"592\" data-end=\"609\">no self-loops<\/strong> and <strong data-start=\"614\" data-end=\"635\">no multiple edges<\/strong>.<\/p>\n<\/li>\n<li data-start=\"637\" data-end=\"724\">\n<p data-start=\"639\" data-end=\"724\"><strong data-start=\"639\" data-end=\"674\">Degree of a Vertex (<span class=\"katex\"><span class=\"katex-mathml\">deg(v)deg(v)<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">d<\/span><span class=\"mord mathnormal\">e<\/span><span class=\"mord mathnormal\">g<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">v<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span>)<\/strong>: The number of edges connected to vertex <span class=\"katex\"><span class=\"katex-mathml\">vv<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">v<\/span><\/span><\/span><\/span>.<\/p>\n<\/li>\n<li data-start=\"725\" data-end=\"971\">\n<p data-start=\"727\" data-end=\"839\"><strong data-start=\"727\" data-end=\"763\">Maximum edges in a Simple Graph:<\/strong> If a simple graph has <strong data-start=\"786\" data-end=\"806\"><span class=\"katex\"><span class=\"katex-mathml\">nn<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">n<\/span><\/span><\/span><\/span> vertices<\/strong>, the maximum number of edges is:<\/p>\n<p><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\">Emax=n(n\u22121)2E_{max} = \\frac{n(n-1)}{2}<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\"><span class=\"mord mathnormal\">E<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">ma<\/span><span class=\"mord mathnormal mtight\">x<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\"><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\">2<span class=\"mord mathnormal\">n<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mbin\">\u2212<\/span>1<span class=\"mclose\">)<\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<p data-start=\"883\" data-end=\"971\">(This occurs when the graph is <strong data-start=\"914\" data-end=\"926\">complete<\/strong>, i.e., every pair of vertices is connected.)<\/p>\n<\/li>\n<\/ul>\n<h3 data-start=\"978\" data-end=\"1010\"><strong data-start=\"982\" data-end=\"1008\">2. Handshaking Theorem<\/strong><\/h3>\n<p data-start=\"1011\" data-end=\"1113\">For any <strong data-start=\"1019\" data-end=\"1039\">undirected graph<\/strong>, the sum of the degrees of all vertices is <strong data-start=\"1083\" data-end=\"1112\">twice the number of edges<\/strong>:<\/p>\n<p><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\">\u2211v\u2208Vdeg(v)=2\u2223E\u2223\\sum_{v \\in V} deg(v) = 2|E|<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mop op-limits\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">v<\/span><span class=\"mrel mtight\">\u2208<\/span><span class=\"mord mathnormal mtight\">V<\/span><\/span><\/span><span class=\"mop op-symbol large-op\">\u2211<\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><span class=\"mord mathnormal\">d<\/span><span class=\"mord mathnormal\">e<\/span><span class=\"mord mathnormal\">g<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">v<\/span><span class=\"mclose\">)<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">2\u2223<\/span><span class=\"mord mathnormal\">E<\/span><span class=\"mord\">\u2223<\/span><\/span><\/span><\/span><\/span><\/p>\n<p data-start=\"1151\" data-end=\"1324\"><strong data-start=\"1154\" data-end=\"1166\">Example:<\/strong><br data-start=\"1166\" data-end=\"1169\" \/>Consider a graph with <strong data-start=\"1191\" data-end=\"1205\">4 vertices<\/strong> and the following degrees:<br data-start=\"1232\" data-end=\"1235\" \/><span class=\"katex\"><span class=\"katex-mathml\">deg(v1)=2deg(v_1) = 2<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">d<\/span><span class=\"mord mathnormal\">e<\/span><span class=\"mord mathnormal\">g<\/span><span class=\"mopen\">(<\/span><span class=\"mord\"><span class=\"mord mathnormal\">v<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">1<\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><\/span><span class=\"mclose\">)<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">2<\/span><\/span><\/span><\/span>, <span class=\"katex\"><span class=\"katex-mathml\">deg(v2)=3deg(v_2) = 3<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">d<\/span><span class=\"mord mathnormal\">e<\/span><span class=\"mord mathnormal\">g<\/span><span class=\"mopen\">(<\/span><span class=\"mord\"><span class=\"mord mathnormal\">v<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><\/span><span class=\"mclose\">)<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">3<\/span><\/span><\/span><\/span>, <span class=\"katex\"><span class=\"katex-mathml\">deg(v3)=3deg(v_3) = 3<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">d<\/span><span class=\"mord mathnormal\">e<\/span><span class=\"mord mathnormal\">g<\/span><span class=\"mopen\">(<\/span><span class=\"mord\"><span class=\"mord mathnormal\">v<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">3<\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><\/span><span class=\"mclose\">)<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">3<\/span><\/span><\/span><\/span>, <span class=\"katex\"><span class=\"katex-mathml\">deg(v4)=2deg(v_4) = 2<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">d<\/span><span class=\"mord mathnormal\">e<\/span><span class=\"mord mathnormal\">g<\/span><span class=\"mopen\">(<\/span><span class=\"mord\"><span class=\"mord mathnormal\">v<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">4<\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><\/span><span class=\"mclose\">)<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">2<\/span><\/span><\/span><\/span>.<br data-start=\"1314\" data-end=\"1317\" \/>Then,<\/p>\n<p><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\">2+3+3+2=10=2\u00d752 + 3 + 3 + 2 = 10 = 2 \\times 5<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\">2<\/span><span class=\"mbin\">+<\/span><\/span><span class=\"base\"><span class=\"mord\">3<\/span><span class=\"mbin\">+<\/span><\/span><span class=\"base\"><span class=\"mord\">3<\/span><span class=\"mbin\">+<\/span><\/span><span class=\"base\"><span class=\"mord\">2<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">10<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">2<\/span><span class=\"mbin\">\u00d7<\/span><\/span><span class=\"base\"><span class=\"mord\">5<\/span><\/span><\/span><\/span><\/span><\/p>\n<p data-start=\"1363\" data-end=\"1393\">So, the graph has <strong data-start=\"1381\" data-end=\"1392\">5 edges<\/strong>.<\/p>\n<h3 data-start=\"1400\" data-end=\"1438\"><strong data-start=\"1404\" data-end=\"1436\">3. Bounds on Number of Edges<\/strong><\/h3>\n<ol data-start=\"1439\" data-end=\"1798\">\n<li data-start=\"1439\" data-end=\"1560\"><strong data-start=\"1442\" data-end=\"1460\">Minimum Edges:<\/strong> A graph with <span class=\"katex\"><span class=\"katex-mathml\">nn<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">n<\/span><\/span><\/span><\/span> vertices can have <strong data-start=\"1500\" data-end=\"1521\">as few as 0 edges<\/strong> (if it&#8217;s an edgeless or null graph).<\/li>\n<li data-start=\"1561\" data-end=\"1798\"><strong data-start=\"1564\" data-end=\"1617\">Maximum Edges in a Simple Graph (Complete Graph):<\/strong>\n<ul data-start=\"1621\" data-end=\"1798\">\n<li data-start=\"1621\" data-end=\"1703\">A <strong data-start=\"1625\" data-end=\"1643\">complete graph<\/strong> <span class=\"katex\"><span class=\"katex-mathml\">KnK_n<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\"><span class=\"mord mathnormal\">K<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">n<\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> has every vertex connected to every other vertex.<\/li>\n<li data-start=\"1707\" data-end=\"1798\">The number of edges in a complete graph is: <span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\">\u2223E\u2223=n(n\u22121)2|E| = \\frac{n(n-1)}{2}<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\">\u2223<\/span><span class=\"mord mathnormal\">E<\/span><span class=\"mord\">\u2223<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\"><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\">2<span class=\"mord mathnormal\">n<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mbin\">\u2212<\/span>1<span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/li>\n<\/ul>\n<\/li>\n<\/ol>\n<h3 data-start=\"1805\" data-end=\"1873\"><strong data-start=\"1809\" data-end=\"1873\">4. Relationship Between Vertices and Edges in Special Graphs<\/strong><\/h3>\n<div class=\"overflow-x-auto contain-inline-size\">\n<table data-start=\"1874\" data-end=\"2233\">\n<thead data-start=\"1874\" data-end=\"1941\">\n<tr data-start=\"1874\" data-end=\"1941\">\n<th data-start=\"1874\" data-end=\"1891\"><strong data-start=\"1876\" data-end=\"1890\">Graph Type<\/strong><\/th>\n<th data-start=\"1891\" data-end=\"1910\"><strong data-start=\"1893\" data-end=\"1909\">Vertices (V)<\/strong><\/th>\n<th data-start=\"1910\" data-end=\"1926\"><strong data-start=\"1912\" data-end=\"1925\">Edges (E)<\/strong><\/th>\n<th data-start=\"1926\" data-end=\"1941\"><strong data-start=\"1928\" data-end=\"1939\">Remarks<\/strong><\/th>\n<\/tr>\n<\/thead>\n<tbody data-start=\"2004\" data-end=\"2233\">\n<tr data-start=\"2004\" data-end=\"2043\">\n<td>Null Graph<\/td>\n<td><span class=\"katex\"><span class=\"katex-mathml\">nn<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">n<\/span><\/span><\/span><\/span><\/td>\n<td>0<\/td>\n<td>No edges<\/td>\n<\/tr>\n<tr data-start=\"2044\" data-end=\"2094\">\n<td>Path Graph<\/td>\n<td><span class=\"katex\"><span class=\"katex-mathml\">nn<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">n<\/span><\/span><\/span><\/span><\/td>\n<td><span class=\"katex\"><span class=\"katex-mathml\">n\u22121n-1<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">n<\/span><span class=\"mbin\">\u2212<\/span><\/span><span class=\"base\"><span class=\"mord\">1<\/span><\/span><\/span><\/span><\/td>\n<td>Single path<\/td>\n<\/tr>\n<tr data-start=\"2095\" data-end=\"2144\">\n<td>Cycle Graph<\/td>\n<td><span class=\"katex\"><span class=\"katex-mathml\">nn<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">n<\/span><\/span><\/span><\/span><\/td>\n<td><span class=\"katex\"><span class=\"katex-mathml\">nn<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">n<\/span><\/span><\/span><\/span><\/td>\n<td>Closed path<\/td>\n<\/tr>\n<tr data-start=\"2145\" data-end=\"2233\">\n<td>Complete Graph <span class=\"katex\"><span class=\"katex-mathml\">KnK_n<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\"><span class=\"mord mathnormal\">K<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">n<\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/td>\n<td><span class=\"katex\"><span class=\"katex-mathml\">nn<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">n<\/span><\/span><\/span><\/span><\/td>\n<td><span class=\"katex\"><span class=\"katex-mathml\">n(n\u22121)2\\frac{n(n-1)}{2}<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\"><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">n<\/span><span class=\"mopen mtight\">(<\/span><span class=\"mord mathnormal mtight\">n<\/span><span class=\"mbin mtight\">\u2212<\/span>1<span class=\"mclose mtight\">)<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/td>\n<td>Maximum edges possible<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<h3 data-start=\"2240\" data-end=\"2275\"><strong data-start=\"2244\" data-end=\"2275\">5. Important GATE Questions<\/strong><\/h3>\n<p data-start=\"2276\" data-end=\"2378\"><strong data-start=\"2279\" data-end=\"2286\">Q1:<\/strong> What is the maximum number of edges in a simple graph with 6 vertices?<br data-start=\"2357\" data-end=\"2360\" \/><strong data-start=\"2363\" data-end=\"2376\">Solution:<\/strong><\/p>\n<p><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\">Emax=6(6\u22121)2=6\u00d752=15E_{max} = \\frac{6(6-1)}{2} = \\frac{6 \\times 5}{2} = 15<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\"><span class=\"mord mathnormal\">E<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">ma<\/span><span class=\"mord mathnormal mtight\">x<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\"><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\">26<span class=\"mopen\">(<\/span>6<span class=\"mbin\">\u2212<\/span>1<span class=\"mclose\">)<\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\"><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\">26<span class=\"mbin\">\u00d7<\/span>5<\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">15<\/span><\/span><\/span><\/span><\/span><\/p>\n<p data-start=\"2440\" data-end=\"2471\">So, the maximum edges = <strong data-start=\"2464\" data-end=\"2470\">15<\/strong>.<\/p>\n<p data-start=\"2473\" data-end=\"2624\"><strong data-start=\"2476\" data-end=\"2483\">Q2:<\/strong> A simple graph has 7 vertices, and the sum of all vertex degrees is 18. Find the number of edges.<br data-start=\"2581\" data-end=\"2584\" \/><strong data-start=\"2587\" data-end=\"2622\">Solution (Handshaking Theorem):<\/strong><\/p>\n<p><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\">\u2211deg(v)=2\u2223E\u2223\\sum deg(v) = 2|E|<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mop op-symbol large-op\">\u2211<\/span><span class=\"mord mathnormal\">d<\/span><span class=\"mord mathnormal\">e<\/span><span class=\"mord mathnormal\">g<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">v<\/span><span class=\"mclose\">)<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">2\u2223<\/span><span class=\"mord mathnormal\">E<\/span><span class=\"mord\">\u2223<\/span><\/span><\/span><\/span><\/span> <span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\">18=2\u2223E\u2223\u21d2\u2223E\u2223=918 = 2|E| \\Rightarrow |E| = 9<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\">18<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">2\u2223<\/span><span class=\"mord mathnormal\">E<\/span><span class=\"mord\">\u2223<\/span><span class=\"mrel\">\u21d2<\/span><\/span><span class=\"base\"><span class=\"mord\">\u2223<\/span><span class=\"mord mathnormal\">E<\/span><span class=\"mord\">\u2223<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">9<\/span><\/span><\/span><\/span><\/span><\/p>\n<p data-start=\"2686\" data-end=\"2716\">So, the graph has <strong data-start=\"2704\" data-end=\"2715\">9 edges<\/strong>.<\/p>\n<h3 data-start=\"2723\" data-end=\"2744\"><strong data-start=\"2727\" data-end=\"2744\">6. Conclusion<\/strong><\/h3>\n<p data-start=\"2745\" data-end=\"2995\">\u00a0A <strong data-start=\"2749\" data-end=\"2765\">simple graph<\/strong> follows basic properties like the <strong data-start=\"2800\" data-end=\"2823\">handshaking theorem<\/strong>.<br data-start=\"2824\" data-end=\"2827\" \/>\u00a0The <strong data-start=\"2833\" data-end=\"2860\">maximum number of edges<\/strong> in a simple graph is <strong data-start=\"2882\" data-end=\"2908\"><span class=\"katex\"><span class=\"katex-mathml\">n(n\u22121)2\\frac{n(n-1)}{2}<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\"><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">n<\/span><span class=\"mopen mtight\">(<\/span><span class=\"mord mathnormal mtight\">n<\/span><span class=\"mbin mtight\">\u2212<\/span>1<span class=\"mclose mtight\">)<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>.<br data-start=\"2909\" data-end=\"2912\" \/>\u00a0These properties help in <strong data-start=\"2939\" data-end=\"2964\">solving GATE problems<\/strong> related to <strong data-start=\"2976\" data-end=\"2992\">graph theory<\/strong>.<\/p>\n<p data-start=\"2997\" data-end=\"3064\" data-is-last-node=\"\" data-is-only-node=\"\">\u00a0<strong data-start=\"3000\" data-end=\"3064\" data-is-last-node=\"\">Need more GATE-level problems on this topic? Let me know!<\/strong><\/p>\n<h3 data-start=\"2997\" data-end=\"3064\"><a href=\"https:\/\/www.maths.ed.ac.uk\/~v1ranick\/papers\/wilsongraph.pdf\" target=\"_blank\" rel=\"noopener\">Discrete mathematics for gate- Relation b\/w Vertices and Edges in Simple graph<\/a><\/h3>\n<h3 class=\"LC20lb MBeuO DKV0Md\"><a href=\"https:\/\/www.zib.de\/userpage\/groetschel\/teaching\/WS1314\/BondyMurtyGTWA.pdf\" target=\"_blank\" rel=\"noopener\">GRAPH THEORY WITH APPLICATIONS<\/a><\/h3>\n<p data-start=\"0\" data-end=\"186\">In <strong data-start=\"3\" data-end=\"27\">Discrete Mathematics<\/strong>, especially in the context of <strong data-start=\"58\" data-end=\"70\">GATE CSE<\/strong>, understanding the <strong data-start=\"90\" data-end=\"151\">relationship between vertices and edges in a simple graph<\/strong> is essential. Let&#8217;s break it down:<\/p>\n<hr data-start=\"188\" data-end=\"191\" \/>\n<h2 data-start=\"193\" data-end=\"227\">\ud83d\udcd8 <strong data-start=\"199\" data-end=\"227\">Definition: Simple Graph<\/strong><\/h2>\n<p data-start=\"229\" data-end=\"280\">A <strong data-start=\"231\" data-end=\"247\">simple graph<\/strong> is an <strong data-start=\"254\" data-end=\"274\">undirected graph<\/strong> that:<\/p>\n<ul data-start=\"281\" data-end=\"421\">\n<li data-start=\"281\" data-end=\"344\">\n<p data-start=\"283\" data-end=\"344\">Has <strong data-start=\"287\" data-end=\"304\">no self-loops<\/strong> (an edge connecting a vertex to itself)<\/p>\n<\/li>\n<li data-start=\"345\" data-end=\"421\">\n<p data-start=\"347\" data-end=\"421\">Has <strong data-start=\"351\" data-end=\"372\">no multiple edges<\/strong> (no two edges connect the same pair of vertices)<\/p>\n<\/li>\n<\/ul>\n<hr data-start=\"423\" data-end=\"426\" \/>\n<h2 data-start=\"428\" data-end=\"483\">\ud83d\udcca <strong data-start=\"434\" data-end=\"483\">Basic Properties \u2013 Vertices (V) and Edges (E)<\/strong><\/h2>\n<p data-start=\"485\" data-end=\"489\">Let:<\/p>\n<ul data-start=\"490\" data-end=\"566\">\n<li data-start=\"490\" data-end=\"529\">\n<p data-start=\"492\" data-end=\"529\"><span class=\"katex\"><span class=\"katex-mathml\">VV<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">V<\/span><\/span><\/span><\/span> be the <strong data-start=\"507\" data-end=\"529\">number of vertices<\/strong><\/p>\n<\/li>\n<li data-start=\"530\" data-end=\"566\">\n<p data-start=\"532\" data-end=\"566\"><span class=\"katex\"><span class=\"katex-mathml\">EE<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">E<\/span><\/span><\/span><\/span> be the <strong data-start=\"547\" data-end=\"566\">number of edges<\/strong><\/p>\n<\/li>\n<\/ul>\n<hr data-start=\"568\" data-end=\"571\" \/>\n<h3 data-start=\"573\" data-end=\"627\">\u2705 <strong data-start=\"579\" data-end=\"627\">1. Maximum Number of Edges in a Simple Graph<\/strong><\/h3>\n<p data-start=\"629\" data-end=\"685\">For a <strong data-start=\"635\" data-end=\"662\">simple undirected graph<\/strong> with <span class=\"katex\"><span class=\"katex-mathml\">nn<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">n<\/span><\/span><\/span><\/span> vertices:<\/p>\n<p><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\">Maximum\u00a0number\u00a0of\u00a0edges,Emax=n(n\u22121)2\\text{Maximum number of edges}, E_{\\text{max}} = \\frac{n(n-1)}{2}<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord text\"><span class=\"mord\">Maximum\u00a0number\u00a0of\u00a0edges<\/span><\/span><span class=\"mpunct\">,<\/span><span class=\"mord\"><span class=\"mord mathnormal\">E<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord text mtight\">max<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\"><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\">2<span class=\"mord mathnormal\">n<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mbin\">\u2212<\/span>1<span class=\"mclose\">)<\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<p data-start=\"760\" data-end=\"840\">\ud83d\udc49 This is because each pair of distinct vertices can have <strong data-start=\"819\" data-end=\"839\">at most one edge<\/strong>.<\/p>\n<hr data-start=\"842\" data-end=\"845\" \/>\n<h3 data-start=\"847\" data-end=\"890\">\u2705 <strong data-start=\"853\" data-end=\"890\">2. Sum of Degrees of All Vertices<\/strong><\/h3>\n<p data-start=\"892\" data-end=\"957\">The <strong data-start=\"896\" data-end=\"906\">degree<\/strong> of a vertex is the number of edges incident on it.<\/p>\n<p><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\">\u2211v\u2208Vdeg(v)=2E\\sum_{v \\in V} \\text{deg}(v) = 2E<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mop op-limits\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">v<\/span><span class=\"mrel mtight\">\u2208<\/span><span class=\"mord mathnormal mtight\">V<\/span><\/span><\/span><span class=\"mop op-symbol large-op\">\u2211<\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><span class=\"mord text\"><span class=\"mord\">deg<\/span><\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">v<\/span><span class=\"mclose\">)<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">2<\/span><span class=\"mord mathnormal\">E<\/span><\/span><\/span><\/span><\/span><\/p>\n<p data-start=\"1000\" data-end=\"1048\">\ud83d\udc49 This is known as the <strong data-start=\"1024\" data-end=\"1047\">Handshaking Theorem<\/strong>.<\/p>\n<hr data-start=\"1050\" data-end=\"1053\" \/>\n<h3 data-start=\"1055\" data-end=\"1093\">\u2705 <strong data-start=\"1061\" data-end=\"1093\">3. Minimum and Maximum Edges<\/strong><\/h3>\n<ul data-start=\"1095\" data-end=\"1252\">\n<li data-start=\"1095\" data-end=\"1190\">\n<p data-start=\"1097\" data-end=\"1135\"><strong data-start=\"1097\" data-end=\"1114\">Minimum edges<\/strong> in a simple graph:<\/p>\n<ul data-start=\"1138\" data-end=\"1190\">\n<li data-start=\"1138\" data-end=\"1190\">\n<p data-start=\"1140\" data-end=\"1190\"><strong data-start=\"1140\" data-end=\"1145\">0<\/strong>, when there are no connections (empty graph)<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<li data-start=\"1192\" data-end=\"1252\">\n<p data-start=\"1194\" data-end=\"1225\"><strong data-start=\"1194\" data-end=\"1211\">Maximum edges<\/strong> (as above):<\/p>\n<ul data-start=\"1228\" data-end=\"1252\">\n<li data-start=\"1228\" data-end=\"1252\">\n<p data-start=\"1230\" data-end=\"1252\"><span class=\"katex\"><span class=\"katex-mathml\">n(n\u22121)2\\frac{n(n-1)}{2}<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\"><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">n<\/span><span class=\"mopen mtight\">(<\/span><span class=\"mord mathnormal mtight\">n<\/span><span class=\"mbin mtight\">\u2212<\/span>1<span class=\"mclose mtight\">)<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<hr data-start=\"1254\" data-end=\"1257\" \/>\n<h2 data-start=\"1259\" data-end=\"1277\">\ud83e\uddee <strong data-start=\"1265\" data-end=\"1277\">Example:<\/strong><\/h2>\n<p data-start=\"1279\" data-end=\"1338\">Let\u2019s say we have a simple graph with <span class=\"katex\"><span class=\"katex-mathml\">n=4n = 4<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">n<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">4<\/span><\/span><\/span><\/span> vertices.<\/p>\n<ul data-start=\"1340\" data-end=\"1566\">\n<li data-start=\"1340\" data-end=\"1432\">\n<p data-start=\"1342\" data-end=\"1365\"><strong data-start=\"1342\" data-end=\"1364\">Max possible edges<\/strong>:<\/p>\n<p><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\">Emax=4(4\u22121)2=122=6E_{\\text{max}} = \\frac{4(4 &#8211; 1)}{2} = \\frac{12}{2} = 6<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\"><span class=\"mord mathnormal\">E<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord text mtight\">max<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\"><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\">24<span class=\"mopen\">(<\/span>4<span class=\"mbin\">\u2212<\/span>1<span class=\"mclose\">)<\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\"><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\">212<\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">6<\/span><\/span><\/span><\/span><\/span><\/li>\n<li data-start=\"1434\" data-end=\"1566\">\n<p data-start=\"1436\" data-end=\"1473\">Suppose we have a graph with 3 edges:<\/p>\n<ul data-start=\"1476\" data-end=\"1566\">\n<li data-start=\"1476\" data-end=\"1513\">\n<p data-start=\"1478\" data-end=\"1513\">Total degree = <span class=\"katex\"><span class=\"katex-mathml\">2\u00d73=62 \\times 3 = 6<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\">2<\/span><span class=\"mbin\">\u00d7<\/span><\/span><span class=\"base\"><span class=\"mord\">3<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">6<\/span><\/span><\/span><\/span><\/p>\n<\/li>\n<li data-start=\"1516\" data-end=\"1566\">\n<p data-start=\"1518\" data-end=\"1566\">So, degrees of all vertices might be: 2, 2, 1, 1<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<hr data-start=\"1568\" data-end=\"1571\" \/>\n<h2 data-start=\"1573\" data-end=\"1612\">\ud83d\udd01 <strong data-start=\"1579\" data-end=\"1612\">Application in GATE Questions<\/strong><\/h2>\n<p data-start=\"1614\" data-end=\"1631\">You may be asked:<\/p>\n<ul data-start=\"1632\" data-end=\"1818\">\n<li data-start=\"1632\" data-end=\"1697\">\n<p data-start=\"1634\" data-end=\"1697\">Find the max\/min number of edges for a given number of vertices<\/p>\n<\/li>\n<li data-start=\"1698\" data-end=\"1747\">\n<p data-start=\"1700\" data-end=\"1747\">Use Handshaking theorem to find missing degrees<\/p>\n<\/li>\n<li data-start=\"1748\" data-end=\"1818\">\n<p data-start=\"1750\" data-end=\"1818\">Identify number of vertices given number of edges and degree pattern<\/p>\n<\/li>\n<\/ul>\n<hr data-start=\"1820\" data-end=\"1823\" \/>\n<h2 data-start=\"1825\" data-end=\"1858\">\ud83e\udde0 <strong data-start=\"1831\" data-end=\"1858\">Key Concepts to Revise:<\/strong><\/h2>\n<ul data-start=\"1860\" data-end=\"2003\">\n<li data-start=\"1860\" data-end=\"1914\">\n<p data-start=\"1862\" data-end=\"1914\">Types of graphs (simple, multigraph, complete, etc.)<\/p>\n<\/li>\n<li data-start=\"1915\" data-end=\"1934\">\n<p data-start=\"1917\" data-end=\"1934\">Handshaking Lemma<\/p>\n<\/li>\n<li data-start=\"1935\" data-end=\"1962\">\n<p data-start=\"1937\" data-end=\"1962\">Complete Graphs <span class=\"katex\"><span class=\"katex-mathml\">KnK_n<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\"><span class=\"mord mathnormal\">K<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">n<\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<\/li>\n<li data-start=\"1963\" data-end=\"2003\">\n<p data-start=\"1965\" data-end=\"2003\">Bipartite Graphs and their edge bounds<\/p>\n<\/li>\n<\/ul>\n<hr data-start=\"2005\" data-end=\"2008\" \/>\n<p data-start=\"2010\" data-end=\"2076\" data-is-last-node=\"\" data-is-only-node=\"\">Would you like a <strong data-start=\"2027\" data-end=\"2061\">diagram or GATE-style question<\/strong> on this topic?<\/p>\n<h3 data-start=\"2010\" data-end=\"2076\"><a href=\"https:\/\/static.pw.live\/5eb393ee95fab7468a79d189\/GLOBAL_CMS_BLOGS\/30aa7d74-582f-4314-ad40-c211dd49d268.pdf\" target=\"_blank\" rel=\"noopener\">Discrete mathematics for gate- Relation b\/w Vertices and Edges in Simple graph<\/a><\/h3>\n<h3 class=\"LC20lb MBeuO DKV0Md\"><a href=\"https:\/\/niamt.ac.in\/WriteReadData\/Mathematics%20(Discrete%20Structure).pdf\" target=\"_blank\" rel=\"noopener\">Mathematics (Discrete Structure).pdf<\/a><\/h3>\n","protected":false},"excerpt":{"rendered":"<p>Discrete mathematics for gate- Relation b\/w Vertices and Edges in Simple graph [fvplayer id=&#8221;160&#8243;] Discrete Mathematics for GATE \u2013 Relationship Between Vertices and Edges in a Simple Graph In graph theory, a simple graph is an unweighted, undirected graph that does not contain multiple edges or self-loops. The relationship between vertices (V) and edges (E) [&hellip;]<\/p>\n","protected":false},"author":71,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[76],"tags":[],"class_list":["post-2890","post","type-post","status-publish","format-standard","hentry","category-discrete-mathematics"],"_links":{"self":[{"href":"https:\/\/www.reilsolar.com\/pdf\/wp-json\/wp\/v2\/posts\/2890","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.reilsolar.com\/pdf\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.reilsolar.com\/pdf\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.reilsolar.com\/pdf\/wp-json\/wp\/v2\/users\/71"}],"replies":[{"embeddable":true,"href":"https:\/\/www.reilsolar.com\/pdf\/wp-json\/wp\/v2\/comments?post=2890"}],"version-history":[{"count":0,"href":"https:\/\/www.reilsolar.com\/pdf\/wp-json\/wp\/v2\/posts\/2890\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.reilsolar.com\/pdf\/wp-json\/wp\/v2\/media?parent=2890"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.reilsolar.com\/pdf\/wp-json\/wp\/v2\/categories?post=2890"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.reilsolar.com\/pdf\/wp-json\/wp\/v2\/tags?post=2890"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}