Matrix Representation of Graph and Digraph

Graph theory plays a fundamental role in various fields of science and engineering, providing powerful tools for modeling and analyzing relationships among entities. One of the most effective ways to study graphs is through matrix representation. This paper explores the three primary matrix representations of graphs: the adjacency matrix, The adjacency matrix provides direct insight into vertex connectivity and the incidence matrix, the incidence matrix reflects the relationship between edges and vertices. and the Laplacian matrix defined as the difference between the degree matrix and the adjacency matrix, plays a central role in spectral graph theory. Matrix representations enable efficient storage, computation, and analysis of graphs using linear algebraic techniques. They form the basis for many modern algorithms in graph theory, This paper discusses the mathematical foundations, construction methods, and practical applications of these matrix forms, highlighting their essential role in both theoretical and applied graph analysis.

Graph Theory, Matrix Representation, Adjacency Matrix

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