{"id":227685,"date":"2025-06-06T11:32:37","date_gmt":"2025-06-06T11:32:37","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=227685"},"modified":"2025-06-06T11:32:39","modified_gmt":"2025-06-06T11:32:39","slug":"consider-the-graph-g-given-below-to-answer-the-following-questions","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/06\/06\/consider-the-graph-g-given-below-to-answer-the-following-questions\/","title":{"rendered":"Consider the graph G given below to answer the following questions."},"content":{"rendered":"\n

Consider the graph G given below to answer the following questions. Find WV. Find d g(el). Find N({d,eh}). Are ad neighbors? How do you know? Is G a multigraph? How do you know? Find path from n to m. Is G bipartite?<\/p>\n\n\n\n

The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n

Thanks for uploading the image of graph GG. Let’s go through each question based on the visual information provided.<\/p>\n\n\n\n

\n\n\n\n

Answers:<\/strong><\/h3>\n\n\n\n

(a) Find EE:<\/strong>
The edge set EE includes the following undirected edges: E={(a,b),(a,e),(a,f),(a,g),(b,c),(b,e),(b,g),(c,d),(c,h),(d,e),(d,h),(e,f),(f,g)}E = \\{(a,b), (a,e), (a,f), (a,g), (b,c), (b,e), (b,g), (c,d), (c,h), (d,e), (d,h), (e,f), (f,g)\\}<\/p>\n\n\n\n

Total: 13 edges<\/strong><\/p>\n\n\n\n

\n\n\n\n

(b) Find \u2223V\u2223|V|:<\/strong>
The vertices are: V={a,b,c,d,e,f,g,h}V = \\{a, b, c, d, e, f, g, h\\}<\/p>\n\n\n\n

Total: 8 vertices<\/strong><\/p>\n\n\n\n

\n\n\n\n

(c) Find deg\u2061(e)\\deg(e):<\/strong>
Edges connected to ee: (a,e), (b,e), (d,e), (e,f)
So, deg\u2061(e)=4\\deg(e) = 4<\/p>\n\n\n\n

\n\n\n\n

(d) Find N({d,e,h})N(\\{d, e, h\\}):<\/strong>
Neighbors of dd: {c, e, h}
Neighbors of ee: {a, b, d, f}
Neighbors of hh: {c, d}
Union (excluding d, e, h): N({d,e,h})={a,b,c,f}N(\\{d, e, h\\}) = \\{a, b, c, f\\}<\/p>\n\n\n\n

\n\n\n\n

(e) Are ee and hh neighbors?<\/strong>
No. There is no edge directly connecting ee and hh.<\/p>\n\n\n\n

\n\n\n\n

(f) Is GG a multigraph?<\/strong>
No. A multigraph has multiple edges between the same two vertices. Here, each edge is unique \u2192 G is not a multigraph.<\/strong><\/p>\n\n\n\n

\n\n\n\n

(g) Path from aa to ee of length 5:<\/strong>
One such path: a\u2192g\u2192f\u2192e\u2192d\u2192ea \\to g \\to f \\to e \\to d \\to e<\/p>\n\n\n\n

This is of length 5, but revisits ee. If simple path required (no repetition), then: a\u2192b\u2192c\u2192h\u2192d\u2192ea \\to b \\to c \\to h \\to d \\to e<\/p>\n\n\n\n

\u2713 Valid, length = 5<\/p>\n\n\n\n

\n\n\n\n

(h) Is G bipartite?<\/strong>
To test bipartiteness, try to 2-color the graph such that no adjacent vertices share a color.
We cannot<\/strong> 2-color because there exists an odd cycle<\/strong>: a\u2192e\u2192f\u2192aa \\to e \\to f \\to a<\/p>\n\n\n\n

Cycle length = 3 \u21d2 G is not bipartite<\/strong><\/p>\n\n\n\n

\n\n\n\n

Explanation:<\/strong><\/h3>\n\n\n\n

Graph theory provides a structured approach to analyze relationships through vertices and edges. In this case, the graph GG contains 8 vertices<\/strong> and 13 undirected edges<\/strong>. Each vertex is connected according to the structure shown in the diagram. For instance, the vertex ee has four edges leading to a,b,d,a, b, d, and ff, giving it a degree of 4<\/strong>.<\/p>\n\n\n\n

When considering the neighborhood of a set of vertices<\/strong> like {d,e,h}\\{d, e, h\\}, we find all vertices directly connected to any in that set, excluding the set itself. Here, the combined neighborhood is {a,b,c,f}\\{a, b, c, f\\}. Regarding adjacency, ee and hh<\/strong> are not directly connected by any edge, so they are not neighbors<\/strong>.<\/p>\n\n\n\n

We also determined that GG is not a multigraph<\/strong>, as no vertex pair is connected by more than one edge. A multigraph would allow multiple parallel edges between the same vertices, which is not observed here.<\/p>\n\n\n\n

In terms of pathfinding, a path of length 5 from aa to ee is possible. One such simple path is a\u2192b\u2192c\u2192h\u2192d\u2192ea \\to b \\to c \\to h \\to d \\to e, involving 5 distinct edges and no repeated vertices.<\/p>\n\n\n\n

Finally, for bipartiteness<\/strong>, we assess whether the graph can be split into two sets where no two vertices in the same set are adjacent. A key property of bipartite graphs is the absence of odd cycles<\/strong>. However, GG contains the cycle a\u2192e\u2192f\u2192aa \\to e \\to f \\to a of length 3, which is odd. Thus, G is not bipartite<\/strong>.<\/p>\n\n\n\n

\"\"<\/figure>\n","protected":false},"excerpt":{"rendered":"

Consider the graph G given below to answer the following questions. Find WV. Find d g(el). Find N({d,eh}). Are ad neighbors? How do you know? Is G a multigraph? How do you know? Find path from n to m. Is G bipartite? The Correct Answer and Explanation is: Thanks for uploading the image of graph […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[25],"tags":[],"class_list":["post-227685","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/227685","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=227685"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/227685\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=227685"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=227685"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=227685"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}