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)= n-ary sumation i=1n|lambda i(L)-d over bar |\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathrm{{LE}}(G) = \sum _{i=1}<^>n|\lambda _i(L)-{\bar{d}}|$$\end{document}, where lambda i(L)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda _i(L)$$\end{document} is the i-th eigenvalue of Laplacian matrix of G, and d over bar \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\bar{d}}$$\end{document} is their average. If LE(G)=LE(Kn)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm{{LE}}(G) = \mathrm{{LE}}(K_n)$$\end{document} for the complete graph Kn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K_n$$\end{document} 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={Gb,j,k=[(((j-2)k-2j+2)b+1)K(j-1)k-(j-2)]boolean OR b(KjxKk)|b,j,k is an element of Z+}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda _1 = \{ G_{b,j,k} = [(((j-2)k-2j+2)b+1)K_{(j-1)k-(j-2)}] \cup b(K_j \times K_k)| b,j,k \in {{\mathbb {Z}}}<^>+\}$$\end{document}, Lambda 2={G2,b=[K6 backward difference b(K2xK3)]boolean OR(4b-2)K9|b is an element of Z+}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \Lambda _2 = \{G_{2,b} = [K_6 \nabla b(K_2 \times K_3)] \cup (4b-2)K_9 | b\in {{\mathbb {Z}}}<^>+ \}$$\end{document}, Lambda 3={G3,b=[bK8 backward difference b(K2xK4)]boolean OR(14b-4)K8b+6|b is an element of Z+}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \Lambda _3 = \{G_{3,b} = [bK_8 \nabla b(K_2 \times K_4)] \cup (14b-4)K_{8b+6} | b\in {{\mathbb {Z}}}<^>+ \}$$\end{document}, where backward difference \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nabla $$\end{document} is join operator and x\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\times $$\end{document} 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={K2 backward difference aK2r over bar |a is an element of Z+}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega _1= \{K_2 \nabla \overline{aK_2<^>r} \vert a\in {{\mathbb {Z}}}<^>+\}$$\end{document}, omega 2={aK3 boolean OR 2(K2xK3) over bar |a is an element of Z+}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega _2 = \{\overline{aK_3 \cup 2(K_2\times K_3)}\vert a\in {{\mathbb {Z}}}<^>+ \}$$\end{document} and omega 3={aK5 boolean OR(K3xK3) over bar |a is an element of Z+}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega _3 = \{\overline{aK_5 \cup (K_3\times K_3)}\vert a\in {{\mathbb {Z}}}<^>+ \}$$\end{document}, where G over bar \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\overline{G}}$$\end{document} is the complement operator on G.
