On the nullity of the line graph of unicyclic graph with depth one

A connected graph with a unique cycle is called a unicyclic graph. A unicyclic graph with depth one may be thought of as being obtained from a cycle by appending ni pendent edges on each vertex vi in the cycle Ct (for some integer t⩾3), denoted by Cn1,n2,…,nt. In this paper, we give a complete characterization on the nullity of the line graph G=L(Cn1,n2,…,nt) as follows: Let . Then(i)η(G)=2 if and only if m=0 and t≡0(mod4).(ii)η(G)=1 if and only if m⩾1 and either(1)ni∈{0,1} for i=1,…,t, the length of any zero chain of (n1,n2,…,nt) is even and t+m≡0(mod4); or(2)t≡0(mod4) and one of n1=n3=⋯=nt-1=0 and n2=n4=⋯=nt=0 must hold.(iii)η(G)=0, otherwise.