GATE CS 2022 Set 1 — Question 52
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Engineering Mathematics → Graphs (Data Structure) → Adjacency Matrix & List
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Question
Which of the properties hold for the adjacency matrix of a simple undirected unweighted graph having vertices?
Correct answer
(A) The diagonal entries of A² are the degrees of the vertices of the graph.
Solution
Option (A): The -th entry of is given by . Since the graph is undirected, is symmetric (), so . For an unweighted simple graph, , so . Thus, . This statement is TRUE.Option (B): Consider a path graph with vertices 1-2-3 (). The adjacency matrix is . Then . . The entries and are zero, yet the graph is connected. This statement is FALSE.Option (C): The sum of all elements of is . The condition implies . A graph with vertices consisting of a triangle () and an isolated vertex has and , satisfying the condition, but it contains a cycle. This statement is FALSE.Option (D): Having at least a 1 in each row/column implies every vertex has a degree of at least 1 (no isolated vertices). Consider a graph with consisting of two disjoint edges (). Every row has a 1, but the graph is disconnected. This statement is FALSE.
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