Get the neighboring vertices and incident edges from a vertex in a graph

A random graph

 g = RandomGraph[{15, 35}, VertexLabels -> "Name", PlotRangePadding -> .2]

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Edges for the vertex 4:

 EdgeList[g, 4 \[UndirectedEdge] _]

Result:

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Now vertexes around vertex4:

 VertexList[NeighborhoodGraph[g, 4]]

Result:

 {4, 14, 11, 6, 7, 9, 13, 15}

Illustration:

  HighlightGraph[g, NeighborhoodGraph[g, 4]]

enter image description here

There are many other ways. You could also use AdjacencyMatrix to get all that info.

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