New families of Laplacian borderenergetic graphs.

Laplacian matrix and its spectrum are commonly used for giving a measure in networks in order to analyse its topological properties. In this paper, Laplacian matrix of graphs and their spectrum are studied. Laplacian energy of a graph G of order n is defined as LE (G) = ∑ i = 1 n | λ i (L) - d ¯ | , where λ i (L) is the i-th eigenvalue of Laplacian matrix of G, and d ¯ is their average. If LE (G) = LE (K n) for the complete graph K n of order n, then G is known as L-borderenergetic graph. In the first part of this paper, we construct three infinite families of non-complete disconnected L-borderenergetic graphs: Λ 1 = { G b , j , k = [ (((j - 2) k - 2 j + 2) b + 1) K (j - 1) k - (j - 2) ] ∪ b (K j × K k) | b , j , k ∈ Z + } , Λ 2 = { G 2 , b = [ K 6 ∇ b (K 2 × K 3) ] ∪ (4 b - 2) K 9 | b ∈ Z + } , Λ 3 = { G 3 , b = [ b K 8 ∇ b (K 2 × K 4) ] ∪ (14 b - 4) K 8 b + 6 | b ∈ Z + } , where ∇ is join operator and × is direct product operator on graphs. Then, in the second part of this work, we construct new infinite families of non-complete connected L-borderenergetic graphs Ω 1 = { K 2 ∇ a K 2 r ¯ | a ∈ Z + } , Ω 2 = { a K 3 ∪ 2 (K 2 × K 3) ¯ | a ∈ Z + } and Ω 3 = { a K 5 ∪ (K 3 × K 3) ¯ | a ∈ Z + } , where G ¯ is the complement operator on G. [ABSTRACT FROM AUTHOR]