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

The nullity of a graph G, denoted by eta(G), is the multiplicity of eigenvalue zero of the adjacency matrix of G. A graph is singular (resp., nonsingular) if eta(G) >= 1 (resp., if eta(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 <= eta(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 eta(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(mod 4) and half the cycles of order equal to 1(mod 4). (C) 2022 Elsevier B.V. All rights reserved.