Recall that a simple graph is a graph such that no vertex is

Recall that a simple graph is a graph such that no vertex is connected to itself by an edge no pair of vertices have more than are edge between them. More ever, a graph is regular of degree d of every vertex has degree d given a simple graph, explain if there is a way to figure out if it regular of degree d from looking at its adjacency matrix Recall the definition of a connected graph show that a simple graph (with at least 2 vertices) is not connected if there are I, j such that I notequalto j such that the (I, j) entry of z A^n is zero for every n greaterthanorequalto 1.

Solution

a) If the sum of each row of the adjacency matrix is equal to d, then it implies that each vertex is connected with d other vertex and the degree of the graph is d

Matrix will be

a11 a12 a13 ...... a1n

a21 a22 a23 ...... a2n

.

.

.

an1 an2 an3 ... ann

Hence if the sum of each row is a constant d, that means the graph is a simple graph with the degree d