Select the correct alternative from the given choices. An...

Select the correct alternative from the given choices. An undirected graph $G$ has ' $n$ ' nodes. Its adjacency matrix is given by an $n \times n$ square matrix. (i) Diagonal elements are 0 's (ii) Non-diagonal elements are l's Which of the following is true? (A) Graph $G$ has no minimum spanning tree (B) Graph $G$ has a unique minimum spanning tree of $\operatorname{cost}(n-1)$ (C) Graph $G$ has multiple distinct minimum spanning trees, each of cost $(n-1)$ (D) Graph $G$ has multiple spanning trees of different cost.

Select the correct alternative from the given choices.
An undirected graph $G$ has ' $n$ ' nodes. Its adjacency matrix is given by an $n \times n$ square matrix.
(i) Diagonal elements are 0 's
(ii) Non-diagonal elements are l's
Which of the following is true?
(A) Graph $G$ has no minimum spanning tree
(B) Graph $G$ has a unique minimum spanning tree of $\operatorname{cost}(n-1)$
(C) Graph $G$ has multiple distinct minimum spanning trees, each of cost $(n-1)$
(D) Graph $G$ has multiple spanning trees of different cost.

Key Concepts

Minimum Spanning Tree (MST)

A Minimum Spanning Tree (MST) is a spanning tree that has the smallest possible total edge weight compared to all other spanning trees in a weighted graph. In cases where every edge has the same weight, as in a complete graph with uniform edge weights, every spanning tree will have the same cost, leading to the existence of multiple distinct minimum spanning trees.

Spanning Tree

A spanning tree of a graph is a subgraph that includes all the vertices of the original graph and is a tree (i.e., it is connected and acyclic). In a graph with n vertices, any spanning tree will have exactly n-1 edges, which is the minimum number of edges required to maintain connectivity without any cycles.

Undirected Graph

An undirected graph is one in which all edges are bidirectional, meaning that there is no distinction between the start and end of an edge. This property implies that the relationships represented by the graph are mutual, leading to symmetric connectivity across its vertices.

Adjacency Matrix

An adjacency matrix is a square matrix used to represent a graph, where the rows and columns correspond to the vertices of the graph. Each entry in the matrix indicates the presence or absence (or weight) of an edge between a pair of vertices. For undirected graphs, this matrix is symmetric, and typically, the diagonal entries are set to zero to indicate that a vertex is not connected to itself.

Complete Graph

A complete graph is a graph in which every pair of distinct vertices is connected by a unique edge. This means that the graph has the maximum number of edges possible without forming loops, and its corresponding adjacency matrix will have all off-diagonal elements as nonzero (or as a constant weight if the graph is weighted), thereby ensuring total interconnectedness among the vertices.

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