Let x be a fixed positive integer. Is it possible to have a graph G with...

Let x be a fixed positive integer. Is it possible to have a graph G with 4x + 1 vertices such that G has a vertex of degree d for all d = 1, 2, ..., 4x + 1? Justify your answer. (Note: The graph G does not need to be simple.)

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Colby Messinger answered 2 years ago

Let G be an abelian group and n a fixed positive integer. Prove that the following...

Let G be an abelian group and n a fixed positive integer. Prove that the following sets are subgroups of G. (a) P(G, n) = {gn | g ∈ G}. (b) T(G, n) = {g ∈ G | gn = 1}. (c) Compute P(G, 2) and T(G, 2) if G = C8 × C2. (d) Prove that T(G, 2) is not a subgroup of G = Dn for n ≥ 3 (i.e the statement above is false when G is...

Let k be an integer satisfying k ≥ 2. Let G be a connected graph with...

Let k be an integer satisfying k ≥ 2. Let G be a connected graph with no cycles and k vertices. Prove that G has at least 2 vertices of degree equal to 1.

If G = (V, E) is a graph and x ∈ V , let G \...

If G = (V, E) is a graph and x ∈ V , let G \ x be the graph whose vertex set is V \ {x} and whose edges are those edges of G that don’t contain x. Show that every connected finite graph G = (V, E) with at least two vertices has at least two vertices x1, x2 ∈ V such that G \ xi is connected.

7. Let m be a fixed positive integer. (a) Prove that no two among the integers...

7. Let m be a fixed positive integer. (a) Prove that no two among the integers 0, 1, 2, . . . , m − 1 are congruent to each other modulo m. (b) Prove that every integer is congruent modulo m to one of 0, 1, 2, . . . , m − 1.

Determine, for a given graph G =V,E and a positive integer m ≤ |V |, whether...

Determine, for a given graph G =V,E and a positive integer m ≤ |V |, whether G contains a clique of size m or more. (A clique of size k in a graph is its complete subgraph of k vertices.) Determine, for a given graph G = V,E and a positive integer m ≤ |V |, whether there is a vertex cover of size m or less for G. (A vertex cover of size k for a graph G =...

CLIQUE INPUT: Graph G, positive integer l PROPERTY: G has a set of l manually adjacent...

CLIQUE INPUT: Graph G, positive integer l PROPERTY: G has a set of l manually adjacent nodes. CLIQUE COVER INPUT: graph G’, positive integer k PROPERTY: N’ is the union of k or fewer cliques. So, Question is : Show that CLIQUE and CLIQUE COVER is cycle base on the property that is given? What does it mean no computer!!

Let A[1..n] be an array of distinct positive integers, and let t be a positive integer....

Let A[1..n] be an array of distinct positive integers, and let t be a positive integer. (a) Assuming that A is sorted, show that in O(n) time it can be decided if A contains two distinct elements x and y such that x + y = t. (b) Use part (a) to show that the following problem, re- ferred to as the 3-Sum problem, can be solved in O(n2) time: 3-Sum Given an array A[1..n] of distinct positive integers, and...

Let a be a positive constant number. Draw the graph of a catenary y=acosh(x/a). Calculate the...

Let a be a positive constant number. Draw the graph of a catenary y=acosh(x/a). Calculate the arc length s from the point (0,a) to the point (x,acosh(x/a)), and find the expression of the curve in terms of the parameter s.

Let G be a group, and let a ∈ G be a fixed element. Define a...

Let G be a group, and let a ∈ G be a fixed element. Define a function Φ : G → G by Φ(x) = ax−1a−1. Prove that Φ is an isomorphism is and only if the group G is abelian.

6. Let t be a positive integer. Show that 1? + 2? + ⋯ + (?...

6. Let t be a positive integer. Show that 1? + 2? + ⋯ + (? − 1)? + ?? is ?(??+1). 7. Arrange the functions ?10, 10?, ? log ? , (log ?)3, ?5 + ?3 + ?2, and ?! in a list so that each function is big-O of the next function. 8. Give a big-O estimate for the function ?(?)=(?3 +?2log?)(log?+1)+(5log?+10)(?3 +1). For the function g in your estimate f(n) is O(g(n)), use a simple function g...

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