On the nullity of a graph with cut-points (Q648933)

Let \(G\) be a graph with the vertex set \(\{v_1,\ldots,v_n\}\). The adjacency matrix of \(G\) is an \(n\times n\) matrix \(A(G)\) whose \((i, j)\)-entry is 1 if \(v_i\) is adjacent to \(v_j\) and 0 otherwise. The nullity of \(G\) is the multiplicity of zero as an eigenvalue of \(A(G)\). It is shown that the set of the nullities of all graphs with exactly \(k\) induced cycles is \(\{0,1,\ldots,k+1\}\). Moreover, it is proved that for a graph \(G\) with \(k\) induced cycles if the nullity of the line graph of \(G\) is \(k+1\), then \(G\) has an even number of vertices.