Table of Contents

- 1 Does every graph has a spanning tree?
- 2 Can all graphs be trees?
- 3 How many minimum spanning trees does a graph have?
- 4 Which is better tree or graph?
- 5 Is minimum spanning tree unique?
- 6 How do you get maximum spanning tree?
- 7 How many spanning trees can a complete graph have?
- 8 What is the definition of a spanning tree?

## Does every graph has a spanning tree?

By this definition, we can draw a conclusion that every connected and undirected Graph G has at least one spanning tree. A disconnected graph does not have any spanning tree, as it cannot be spanned to all its vertices.

**How do you tell if a graph has a spanning tree?**

Theorem: A graph is connected iff it has a spanning tree. Proof: If a graph is connected, we can identify a cycle and remove an edge from it: it will still be connected. We can continue this until no cycles remain. The result is a spanning tree.

### Can all graphs be trees?

Since a tree contains no cycles at all, it is bipartite. Every tree is a median graph. Every tree with only countably many vertices is a planar graph. Every connected graph G admits a spanning tree, which is a tree that contains every vertex of G and whose edges are edges of G.

**Are all trees spanning?**

A spanning tree is a tree (as per the definition in the question) that is spanning. is also not a spanning tree (it’s spanning, but it’s not a tree). A tree is just a type of graph (connected and no cycles). Saying it is a tree is simpler and has less information.

## How many minimum spanning trees does a graph have?

one minimum spanning tree

There is only one minimum spanning tree in the graph where the weights of vertices are different.

**What is maximum spanning tree?**

A maximum spanning tree is a spanning tree of a weighted graph having maximum weight. It can be computed by negating the weights for each edge and applying Kruskal’s algorithm (Pemmaraju and Skiena, 2003, p. 336). A maximum spanning tree can be found in the Wolfram Language using the command FindSpanningTree[g].

### Which is better tree or graph?

Vertices are nothing but the nodes in the graph. Two adjacent vertices are joined by edges….Graph vs Tree.

No. | Graph | Tree |
---|---|---|

1 | Graph is a non-linear data structure. | Tree is a non-linear data structure. |

2 | It is a collection of vertices/nodes and edges. | It is a collection of nodes and edges. |

**Why Every tree is a graph but not every graph is a tree?**

Answer: Every tree is a bipartite graph. Since a tree contains no cycles at all, it is bipartite. Every connected graph G admits a spanning tree, which is a tree that contains every vertex of G and whose edges are edges of G.

## Is minimum spanning tree unique?

If the edge weights are all positive, it suffices to define the MST as the subgraph with minimal total weight that connects all the vertices. The edge weights are all different. If edges can have equal weights, the minimum spanning tree may not be unique.

**Which is better Prims or Kruskal?**

Prim’s algorithm is significantly faster in the limit when you’ve got a really dense graph with many more edges than vertices. Kruskal performs better in typical situations (sparse graphs) because it uses simpler data structures.

### How do you get maximum spanning tree?

**What is minimum and maximum spanning tree?**

The degree constrained minimum spanning tree is a minimum spanning tree in which each vertex is connected to no more than d other vertices, for some given number d. A maximum spanning tree is a spanning tree with weight greater than or equal to the weight of every other spanning tree.

## How many spanning trees can a complete graph have?

Spanning tree has n-1 edges, where n is the number of nodes (vertices). From a complete graph, by removing maximum e – n + 1 edges, we can construct a spanning tree. A complete graph can have maximum n n-2 number of spanning trees.

**Can a spanning tree have the same number of vertices?**

All the possible spanning trees that can be created from the given graph G would have the same number of vertices, but the number of edges in the spanning tree would be equal to the number of vertices in the given graph minus 1. The spanning tree does not contain any cycle. Let’s understand this property through an example.

### What is the definition of a spanning tree?

A spanning tree is a subset of Graph G, which has all the vertices covered with minimum possible number of edges. Hence, a spanning tree does not have cycles and it cannot be disconnected.. By this definition, we can draw a conclusion that every connected and undirected Graph G has at least one spanning tree.

**How is the weight of a minimum spanning tree measured?**

In a weighted graph, a minimum spanning tree is a spanning tree that has minimum weight than all other spanning trees of the same graph. In real-world situations, this weight can be measured as distance, congestion, traffic load or any arbitrary value denoted to the edges.