planar gr

17. Let A and B be two n × n matrices. Show that a) (A + B)^t = A^t + B^t . b) (AB)^t = B^t A^t . If A and B are n × n matrices with AB = BA = In, then B is called the inverse of A (this terminology is appropriate because such a matrix B is unique) and A is said to be invertible. The notation B = A^(−1) denotes that B is the inverse of A.

. For each of the following, draw an Eulerian graph that satisfies the conditions, orprove that no such graph exists.a) An even number of vertices, an even number of edges.b) An even number of vertices, an odd number of edges.c) An odd number of vertices, an even number of edges.d) An odd number of vertices, an odd number of edges.

(a) Show that a graph G with at least three vertices is 2-connected if and only if any vertex and any edge of G lie on a common cycle of G: (b) Show that a graph G with at least three vertices is 2-connected if and only if any two edges of G lie on a common cycle

Applied combinatorics   A directed graph is called strongly connected if there is a directed path fromany given vertex to any other vertex. Show that if a directed graph possesses a directed Euler cycle, then it must be strongly connected.

What is the minimum number of 1’s in the adjacency matrix of a connected undirected graph of N vertices (without self–loops) ? How many 1’s are in the adjacency matrix of a complete graph of N vertices?

Given a graph G, please do the following: a) Show the adjacency matrix for graph G. b) Show the adjacency list for graph G. I 1 6 2 4 7 3 5 Graph G

Consider the following graph: What is the chromatic number of this graph?

Number Guessing Game.  The program will generate a random number from 1 to 1000.  The player will be given several chances to guess the number.  The program gives a hint that the given number is lower or higher. If the player's guess is correct, give the player a single point. Tally the number of correct and wrong guesses that the player has made. After every round, display the tally (correct, wrong, and score)  After a cycle the program will ask the user if he wants to continue playing. This is a console project.   Below source code is my answer with the Question. Kindly check it sir/maam if i have the problem with the code. Thank you using System;using System.Collections.Generic;using System.Text;class Program{static void Main(string[] args){while (true){int randno = Newnum(1, 1001);int count = 1;while (true){Console.Write("Enter a number between 1 and 1000(0 to quit):");int input = Convert.ToInt32(Console.ReadLine());if (input == 0)return;else if (input <…

The following conditions must be met for a graph to include an Euler circuit: (1) it must be linked, and (2) each vertex must have an even degree. What are the conditions for having an Euler path?