Finding a Minimum Spanning Tree using QGIS

There is an already existing tool in processing plugin in GRASS -> Vector ->

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You will need to specify the vector line layer and the nodes ( optional ) and in advanced parameters add the type of the geometry of the output layer

and that's it, you'll get a new vector layer with the spanning tree

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You may also rely on the 'Minimum Spanning Tree' plugin of QGIS 3.x.

The paper Geo-MST: A geographical minimum spanning tree plugin for QGIS discusses the plugin thoroughly.