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The adjacency matrix is used in discrete mathematics to represent the number of ways in which we can walk from one vertex to another within a graph. Any graph can be shown in an adjacency matrix where both the rows and columns are labeled with our graph vertices. We denote each entry as $$\left(i,j\right)$$ which counts the number of adjacent edges between the $$i^{th}$$ row and $$j^{th}$$ column. We also say that $$a_{i,j}$$ represents the number in row $$i$$, column $$j.$$ The adjacency matrix is made up of graph vertices that are either a $$0$$ or $$1.$$ To decide which entry to write in the matrix, we use a $$0$$ if vertex $$i$$ is not adjacent to vertex $$j$$ and we use a $$1$$ if vertex $$i$$ is adjacent to vertex $$j.$$
Notice, that both vertices $$1$$ and $$3$$ are each adjacent to vertices $$2$$ and $$4.$$ Vertices $$2$$ and $$4$$ are each adjacent to vertices $$1$$ and $$3.$$ With that being said, we can write out our adjacency matrix as follows:
Notice that our adjacency matrix is symmetric with $$0$$'s on the diagonal. All adjacency matrices will be adjacent if the given graph in undirected.