In the first part of this paper, we present the Laplacian spectrum of some graphs composed of complete graphs. To begin with, we give the Laplacian spectrum of G1 = aKi ∪ b(Kj × Kk) and then we find the values of i, b, j, k ∈ Z + for which G1 is Laplacian borderenergetic. Similarly, we give the Laplacian spectrum of G2 = aKi∇b(Kj × Kk) ∪ cKl and find the values of a, i, b, j, k, c, l ∈ Z + for which G2 is Laplacian borderenergetic. In this part, we construct three infinite families of non-complete disconnected L-borderenergetic graphs: Λ1 = {Gb,j,k = (((j − 2)k − 2j + 2)b + 1)K (j−1)k−(j−2)) ∪ b(Kj × Kk)|b, j, k ∈ Z + }, Λ2 = {G2,b = [K6∇b(K2 × K3)] ∪ (4b − 2)K9|b ∈ Z + }, Λ3 = {G3,b = [bK8∇b(K2 × K4)] ∪ (14b − 4)K8b+6|b ∈ Z + }. Then, in the second part of this work, we construct new infinite families of non-complete connected L-borderenergetic graphs Ω1 = {K2∇aK r 2 |a ∈ Z + }, Ω2 = {aK3 ∪ 2(K2 × K3)|a ∈ Z + } and Ω3 = {aK5 ∪ (K3 × K3)|a ∈ Z + }.
MSC Classification: 05C50 , 05C76 , 15A18