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3. Show that the sum of the degrees of the vertices of the graph G is equal to twice the number of edges, that is Σd(v) = 2-|E(G)|. VEV (G) Remember that a loop contributes 2 to the degree of the vertex where it is located.

3. Show that the sum of the degrees of the vertices of the graph G is equal to twice the number of edges, that is Σd(v) = 2-|E(G)|. VEV (G) Remember that a loop contributes 2 to the degree of the vertex where it is located.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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3. Show that the sum of the degrees of the vertices of the graph G is equal to twice the number of edges, that is Σd(v) = 2. |E(G). VEV (G) Remember that a loop contributes 2 to the degree of the vertex where it is located. 4. Let G = (V, E) and G' = (V', E') be simple graphs. We say that a graph G is homomorphic to a graph G' if there exists a mapping ø: V → V' such that for each edge {x,y} E of the graph G it follows that {o(r), (y)} = E'. Such a mapping o is called a homomorphism of the graph G onto the graph G', and is denoted by : G → G'. Show that for any three graphs G₁, G2, G3, if there exists a homomorphism f: G₁ → G₂ and a homomorphism g: G₂ → G3, then there also exists a homomorphism h : G₁ → G3. Using the proof, then find the corresponding homomorphisms for the following graphs: P y 9 G₁ "1 21 t Q 9 C G3 b G₂ 5. There are two special types of vertices of interest for digraphs. A vertex that is not an initial vertex (tail) of any edge, i.e., one with no "arrows" leading away from it, is called a sink. A source is a vertex that is not a terminal vertex (head) of any edge. Show that every acyclic digraph has at least one sink and at least one source.
Transcribed Image Text:3. Show that the sum of the degrees of the vertices of the graph G is equal to twice the number of edges, that is Σd(v) = 2. |E(G). VEV (G) Remember that a loop contributes 2 to the degree of the vertex where it is located. 4. Let G = (V, E) and G' = (V', E') be simple graphs. We say that a graph G is homomorphic to a graph G' if there exists a mapping ø: V → V' such that for each edge {x,y} E of the graph G it follows that {o(r), (y)} = E'. Such a mapping o is called a homomorphism of the graph G onto the graph G', and is denoted by : G → G'. Show that for any three graphs G₁, G2, G3, if there exists a homomorphism f: G₁ → G₂ and a homomorphism g: G₂ → G3, then there also exists a homomorphism h : G₁ → G3. Using the proof, then find the corresponding homomorphisms for the following graphs: P y 9 G₁ "1 21 t Q 9 C G3 b G₂ 5. There are two special types of vertices of interest for digraphs. A vertex that is not an initial vertex (tail) of any edge, i.e., one with no "arrows" leading away from it, is called a sink. A source is a vertex that is not a terminal vertex (head) of any edge. Show that every acyclic digraph has at least one sink and at least one source.
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