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<\/figure>\n\n\n\nAn adjacency matrix is a simple and structured way to represent a graph in programming<\/a>. It is a two-dimensional array (N \u00d7 N), where N is the total number of vertices (or nodes) in the graph. Each row and column of this matrix represents a vertex, and the value stored at any cell shows whether there is a connection (edge) between two vertices.<\/p>\n\n\n\n In simple words, an adjacency matrix tells us which nodes are connected to which, and how they are connected. This makes it one of the easiest ways to store and visualize relationships in a graph.<\/p>\n\n\n\n Here\u2019s how it works:<\/p>\n\n\n\n The main advantage of using an adjacency matrix is that it allows you to quickly check if an edge exists between two nodes in constant time O(1). This makes it highly efficient for operations where fast edge lookup is required, such as in graph traversal or shortest-path algorithms.<\/p>\n\n\n\n In short, an adjacency matrix in DSA<\/a> is like a table that keeps track of all possible connections in a graph \u2014 making it easier to perform computations, analyze structures, and build algorithms efficiently.<\/p>\n\n\n\n An adjacency matrix is one of the most efficient ways to represent a graph when you need quick access to connections between nodes. It provides a clear, table-like structure that allows fast and direct checking of whether an edge exists between any two vertices.<\/p>\n\n\n\n You should use an adjacency matrix in the following situations:<\/p>\n\n\n\n However, there\u2019s one major drawback: an adjacency matrix always takes up O(N\u00b2) space. This means that even if your graph has very few connections (a sparse graph), the matrix still occupies memory for every possible pair of vertices. So while it\u2019s excellent for dense graphs and quick lookups, it\u2019s not memory-efficient for large or sparse graphs.<\/p>\n\n\n\n To truly understand why and when to use an adjacency matrix, learning the core data structures and algorithms behind it is essential. HCL GUVI\u2019s DSA eBook<\/a> helps you master these basics \u2014 from arrays and matrices to graphs and dynamic programming \u2014 with clear explanations, real examples, and practice questions. It\u2019s an ideal companion for beginners and interview aspirants who want to build strong graph fundamentals.<\/p>\n\n\n\n An adjacency matrix is one of the most popular and easy-to-understand ways to represent a graph in programming. Whether you are working with a simple unweighted graph or a complex weighted and directed one, understanding how to create, update, and maintain an adjacency matrix is an essential DSA skill.<\/p>\n\n\n\n Let\u2019s break down the entire process step by step using simple terms, clear explanations, and practical code examples.<\/p>\n\n\n\n The first step in creating an adjacency matrix is initializing the structure<\/strong>.<\/p>\n\n\n\n If a graph has N vertices<\/strong>, you\u2019ll need an N \u00d7 N matrix<\/strong> (a two-dimensional array) where each cell represents a possible connection between two nodes.<\/p>\n\n\n\n This setup makes it easy to modify and check connections later.<\/p>\n\n\n\n Once your matrix is initialized, you can add an edge<\/strong> between two vertices.<\/p>\n\n\n\n Adding an edge takes O(1)<\/strong> time \u2014 meaning it\u2019s instant, regardless of how large your graph is.<\/p>\n\n\n\n To remove an edge<\/strong>, simply reset the corresponding entries in the matrix:<\/p>\n\n\n\n This operation also takes O(1)<\/strong> time.<\/p>\n\n\n\n The biggest advantage of an adjacency matrix is how fast you can check whether an edge exists between two nodes.<\/p>\n\n\n\n You just look at the cell [u][v]:<\/p>\n\n\n\n This check happens in constant time O(1)<\/strong> \u2014 one of the key reasons adjacency matrices are used in graph algorithms that require frequent lookups.<\/p>\n\n\n\n To find all the neighbors (connected vertices) of a particular node, you can scan its entire row in the matrix.<\/p>\n\n\n\n For example, to find all nodes connected to vertex u, check every column in row u and collect the indices where a connection exists (matrix[u][v] == 1).<\/p>\n\n\n\n This takes O(N)<\/strong> time per vertex because you have to check every possible node once.<\/p>\n\n\n\n Before jumping into the code, let\u2019s understand what unweighted and weighted graphs mean.<\/p>\n\n\n\n 1. Python code for unweighted graph<\/strong><\/p>\n\n\n\n This code demonstrates how to add, remove, and check edges in a simple graph structure.<\/p>\n\n\n\n 2. Python code for weighted graph<\/strong><\/p>\n\n\n\n If your graph has weights and directions, you can use the following version:<\/p>\n\n\n\n\n
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<\/li>\n\n\n\nWhy Use an Adjacency Matrix?<\/strong><\/h1>\n\n\n\n
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<\/strong>If you often need to verify whether two nodes are connected, an adjacency matrix is perfect. You can find this information instantly in O(1) time because each cell directly stores the connection status.
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<\/strong>A graph is called dense when it has many edges \u2014 close to the total possible number of edges (N\u00b2). In such cases, using an adjacency matrix makes sense because most cells in the matrix will contain valid connections, making the memory usage worthwhile.
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<\/strong>Some algorithms, like the Floyd\u2013Warshall algorithm (used to find shortest paths between all pairs of vertices), perform operations on every pair of nodes. The adjacency matrix works best here because it allows immediate access to every connection without searching through lists.<\/li>\n<\/ul>\n\n\n\nHow to Create and Maintain an Adjacency Matrix\u00a0<\/strong><\/h1>\n\n\n\n
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<\/figure>\n\n\n\n1. Initialization<\/strong><\/h2>\n\n\n\n
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<\/strong> You can initialize every cell with 0, meaning \u201cno edge exists.\u201d
When you add an edge between two nodes, you\u2019ll change the corresponding entries to 1.
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<\/strong> You usually initialize every cell with a large number like INF (infinity) or sometimes -1 to indicate \u201cno connection.\u201d
If the graph allows self-loops, set the diagonal entries (where a node connects to itself) to 0 since the distance from a node to itself is zero.<\/li>\n<\/ul>\n\n\n\n2. Adding an Edge<\/strong><\/h2>\n\n\n\n
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<\/strong> Since the connection goes both ways, you must update both [u][v] and [v][u] entries to 1 (or to the edge weight in case of weighted graphs).<\/li>\n\n\n\n
<\/strong> The edge goes in one direction, so you only update [u][v] and leave [v][u] unchanged.<\/li>\n<\/ul>\n\n\n\n3. Removing an Edge<\/strong><\/h2>\n\n\n\n
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4. Checking if an Edge Exists<\/strong><\/h2>\n\n\n\n
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5. Iterating Through Neighbors<\/strong><\/h2>\n\n\n\n
6. Python Example <\/strong><\/h2>\n\n\n\n
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<\/figure>\n\n\n\n# Adjacency matrix for unweighted undirected graph\nN = 5\nadj = [[0] * N for _ in range(N)]\n\ndef add_edge(u, v):\n adj[u][v] = 1\n adj[v][u] = 1 # undirected graph\n\ndef remove_edge(u, v):\n adj[u][v] = 0\n adj[v][u] = 0\n\ndef has_edge(u, v):\n return adj[u][v] == 1\n\n# Example usage\nadd_edge(0, 1)\nadd_edge(1, 2)\nprint(has_edge(0, 1)) # True\nprint(adj)\n<\/code><\/pre>\n\n\n\n![\"](\"https:\/\/www.guvi.in\/blog\/wp-content\/uploads\/2026\/01\/5-1-1.png\" "\\"\\"")
<\/figure>\n\n\n\nINF = 10**9 # a large number representing 'no edge'\nN = 4\nadj = [[INF] * N for _ in range(N)]\n\nfor i in range(N):\n adj[i][i] = 0 # distance from a node to itself\n\ndef add_edge(u, v, w):\n adj[u][v] = w # directed edge from u to v with weight w\n\n# Example edges\nadd_edge(0, 1, 5)\nadd_edge(1, 2, 3)\n\n# Example: adj[0][1] == 5, adj[1][0] == INF\n<\/code><\/pre>\n\n\n\n
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