Laplacian Spectrum of Graphs and new L-Borderenergetic Families

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