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 \( \mathrm{{LE}}(G) = \sum _{i=1}^n|\lambda _i(L)-{\bar{d}}|\), where \(\lambda _i(L)\) is the i-th eigenvalue of Laplacian matrix of G, and \({\bar{d}}\) is their average. If \(\mathrm{{LE}}(G) = \mathrm{{LE}}(K_n)\) 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: \(\Lambda _1 = \{ G_{b,j,k} = [(((j-2)k-2j+2)b+1)K_{(j-1)k-(j-2)}] \cup b(K_j \times K_k)| b,j,k \in {{\mathbb {Z}}}^+\}\), \( \Lambda _2 = \{G_{2,b} = [K_6 \nabla b(K_2 \times K_3)] \cup (4b-2)K_9 | b\in {{\mathbb {Z}}}^+ \}\), \( \Lambda _3 = \{G_{3,b} = [bK_8 \nabla b(K_2 \times K_4)] \cup (14b-4)K_{8b+6} | b\in {{\mathbb {Z}}}^+ \}\), where \(\nabla \) is join operator and \(\times \) 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 \(\Omega _1= \{K_2 \nabla \overline{aK_2^r} \vert a\in {{\mathbb {Z}}}^+\}\), \(\Omega _2 = \{\overline{aK_3 \cup 2(K_2\times K_3)}\vert a\in {{\mathbb {Z}}}^+ \}\) and \(\Omega _3 = \{\overline{aK_5 \cup (K_3\times K_3)}\vert a\in {{\mathbb {Z}}}^+ \}\), where \({\overline{G}}\) is the complement operator on G.