Assume a graph G is simple (ie. no self loop or parallel edges) . Let v be any vertex in the graph.   Let boolean[]  marked be initalized to all false.   Consider:

Database System Concepts
7th Edition
ISBN:9780078022159
Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Chapter1: Introduction
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Assume a graph G is simple (ie. no self loop or parallel edges) .

Let v be any vertex in the graph.  
Let boolean[]  marked be initalized to all false.
 
Consider:
   boolean dfs(Graph G, int v) {
        marked[v] = true;
        for (int w : G.adj(v)) {
            if (w == u) continue; 
            if (marked[w])  return true;
            if (dfs (G, v, w)) return true;
        }
        return false;
    }
  
If the call dfs(v) returns true, then:
 
a.  The graph has cycles
b.  The graph is bipartite
c.  The graph is connected
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