ii. Let G = (V, E) be a tree. Prove G has |V | − 1...

ii. Let G = (V, E) be a tree. Prove G has |V | − 1 edges using strong induction. Hint: In the inductive step, choose an edge (u, v) and partition the set vertices into two subtrees, those that are reachable from u without traversing (u, v) and those that are reachable from v without traversing (u, v). You will have to reason why these subtrees are distinct subgraphs of G.

iii. What is the total degree of a tree?

Expert Solution

Colby Messinger answered 2 years ago

Let G be connected, and let e be an edge of G. Prove that e is...

Let G be connected, and let e be an edge of G. Prove that e is a bridge if and only if it is in every spanning tree of G.

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.

Let T = (V,E) be a tree, and letr, r′ ∈ V be any two nodes....

Let T = (V,E) be a tree, and letr, r′ ∈ V be any two nodes. Prove that the height of the rooted tree (T, r) is at most twice the height of the rooted tree (T, r′).

Prove or disprove: If G = (V; E) is an undirected graph where every vertex has...

Prove or disprove: If G = (V; E) is an undirected graph where every vertex has degree at least 4 and u is in V , then there are at least 64 distinct paths in G that start at u.

Let G = (V, E) be a directed graph, with source s ∈ V, sink t...

Let G = (V, E) be a directed graph, with source s ∈ V, sink t ∈ V, and nonnegative edge capacities {ce}. Give a polynomial-time algorithm to decide whether G has a unique minimum s-t cut (i.e., an s-t of capacity strictly less than that of all other s-t cuts).

Let G be a graph. prove G has a Eulerian trail if and only if G...

Let G be a graph. prove G has a Eulerian trail if and only if G has at most one non-trivial component and at most 2 vertices have odd degree

Prove 1. Let f : A→ B and g : B → C . If g...

Prove 1. Let f : A→ B and g : B → C . If g 。 f is one-to-one, then f is one-to-one. 2. Equivalence of sets is an equivalence relation (you may use other theorems without stating them for this one).

1) a) Let k ≥ 2 and let G be a k-regular bipartite graph. Prove that G...

1) a) Let k ≥ 2 and let G be a k-regular bipartite graph. Prove that G has no cut-edge. (Hint: Use the bipartite version of handshaking.) b) Construct a simple, connected, nonbipartite 3-regular graph with a cut-edge. (This shows that the condition “bipartite” really is necessary in (a).) 2) Let F_n be a fan graph and Let a_n = τ(F_n) where τ(F_n) is the number of spanning trees in F_n. Use deletion/contraction to prove that a_n = 3a_n-1 - a_n-2...

Let G be a nontrivial nilpotent group. Prove that G has nontrivial center.

Let A be incidence matrix of graph G. prove if G has a cycle, then null...

Let A be incidence matrix of graph G. prove if G has a cycle, then null space of transpose of A is not {0} (there exists non-trivial solution of (A^T)y=0)

Question