Nullity and singularity of a graph in which every block is a cycle

The nullity of a graph G, denoted by η(G), is the multiplicity of eigenvalue zero of the adjacency matrix of G. A graph is singular (resp., nonsingular) if η(G)≥1 (resp., if η(G)=0). A cycle-spliced graph is a cactus in which every block is a cycle. Recently, Singh et al. [16] consider the singularity of graphs in which every block is a clique. In this paper, we consider the nullity and the singularity of cycle-spliced graphs. Let G be a cycle-spliced graph with c(G) cycles. If G is bipartite, we prove that 0≤η(G)≤c(G)+1, the extremal graphs G with nullity 0 or c(G)+1 are respectively characterized. If all cycles in G are odd, we obtain the following two results: (i) G is nonsingular if c(G) is odd, and η(G) is 0 or 1 if c(G) is even. (ii) If every cycle in G has at most two cut-vertices of G, then G is singular if and only if c(G) is even and G contains half the cycles of order equal to 3(mod4) and half the cycles of order equal to 1(mod4).