molecular structure descriptors or topological indices are used for modelling information of molecules , including toxicologic , chemical , and other properties of chemical compounds in theoretical chemistry . topological indices play a very important role in mathematical chemistry , especially in the quantitative structure - property relationship ( qspr ) and quantitative structure activity relationship ( qsar ) . many topological indices have been introduced and investigated by mathematicians , chemists , and biologists , which contain energy , the laplacian - energy - like invariant [ 25 ] , the kirchhoff index [ 613 ] , and so forth . the energy of the graph is an important invariant of the adjacency spectrum and is the sum of the absolute values of all the eigenvalues of a graph g , which is studied in chemistry and used to approximate the total electron energy of a molecule . during researching the character of the conjugated carbon oxides e 
  is closely related to the energy releasing from the formation progress of the conjugated carbon oxides and could be approximately calculated by hckel molecular orbital theory . and in the method of hmo , the calculation of e 
  can be attributed to the sum of the absolute values of all the eigenvalues of its molecular graph [ 1420 ] . compared with adjacency matrix , the definition of laplacian matrix added to all vertices degrees . as mohar said , cvetkovi and simi [ 2123 ] pointed out that , as molecular structure descriptors , the laplacian - energy - like invariant not only well describes the properties of most of the descriptors which are indicated , such as entropy , molar volume , and molar refractivity , but also is able to describe some more difficult properties , such as boiling point and rub points . due to the fact that laplacian - energy - like invariant has a significant physical and chemical background [ 24 , 25 ] , it has received wide attention to research it from many mathematical and chemical workers . all the graphs discussed in this paper are simple , finite , and undirected . for a graph g , the vertex set and edge set of g will be denoted by v(g ) = { v 
 1 , v 
 2 ,  , v 
 n } and e(g ) = { e 
 1 , e 
 2 , , e 
 m } , respectively . the adjacency matrix and the diagonal matrix of g are , respectively , a(g ) and d(g ) ; then the matrix l(g ) = d(g )  a(g ) is called the laplacian matrix of the graph g [ 27 , 28 ] . the characteristic polynomials and laplacian polynomials of the graph g are  
 g( ) = det(i  a(g ) ) and  
 g( ) = det(i  l(g ) ) . both a(g ) and l(g ) are symmetric matrices ; their eigenvalues are real numbers [ 30 , 31 ] . thus , we can order the eigenvalues of the graph g as  
 1   
 2   
 n , and the laplacian eigenvalues are  
 1   
 2   
 n [ 32 , 33 ] . if g is a connected graph , then  
 i > 0 , i = 1,2 ,  , n  1 ,  
 n = 0  [ 3436 ] . the energy of a graph g is the sum of the absolute values of all the eigenvalues of g ; that is,(1)eg=i=1ni . the energy of a graph g is the sum of the absolute values of all the eigenvalues of g ; that is,(1)eg=i=1ni . let g be a graph of order n. the laplacian - energy - like invariant of g , denoted by lel(g ) , is defined as(2)lelg=i=1ni . let g be a graph of order n. the laplacian - energy - like invariant of g , denoted by lel(g ) , is defined as(2)lelg=i=1ni . definition 3 ( see ) . for two matrices a = ( a 
 i , j)mn , b = ( b 
 i , j)st , the tensor product of a and b , denoted by a  b , is defined as(3)a11ba12ba1nbam1bam2bamnb . 

  for two matrices a = ( a 
 i , j)mn , b = ( b 
 i , j)st , the tensor product of a and b , denoted by a  b , is defined as(3)a11ba12ba1nbam1bam2bamnb . let { g 
 n } be a sequence of finite simple graphs with bounded average degree such that ( 4)limnvgn=,limnlelgnvgn = h0 . n } be a sequence of spanning subgraphs of { g 
 n } such that ( 5)limnvvhn : dhnv = dgnvvgn=1;then ( 6)limnlelhnvgn = h . 
 that is , g 
 n and h 
 n have the same asymptotic laplacian - energy - like invariant . let { g 
 n } be a sequence of finite simple graphs with bounded average degree such that ( 4)limnvgn=,limnlelgnvgn = h0 . n } be a sequence of spanning subgraphs of { g 
 n } such that ( 5)limnvvhn : dhnv = dgnvvgn=1;then ( 6)limnlelhnvgn = h . that is , g 
 n and h 
 n have the same asymptotic laplacian - energy - like invariant . in what follows , we will explore the laplacian - energy - like invariants formulas of the modified hexagonal lattice , modified union jack lattice , and honeycomb lattice . the modified hexagon lattice with toroidal boundary condition is denoted by mh(n 
 1 , n 
 2 ) . let  
 i = 2i / n 
 1 ,  
 j = 2j / n 
 2 . i = 2i / n 
 1 ,  
 j = 2j / n 
 2 . proofwith the proper labelling of the vertices of the modified hexagonal lattice , its laplacian matrix is 
 ( 8)where i 
 n1 , i 
 n2 are the unit matrices and m  n is tensor product of matrices m and n. consider(9)bn2=01000001000000110000n2n2 . 
 the matrix l(mh(n 
 1 , n 
 2 ) ) can be defined as follows : ( 10)lmhtn1,n2in2cn1bn2pn1bn2tpn1t = in26in1bn1bn1tbn2in1+bn1tbn2tin1+bn1 . , g 
 } be a cyclic group of order n. obviously ,  : g 
   b 
 n 
  can express the group . the cyclic group of order n has linear values of n   
 i  ( i = 0,1 ,  , n  1 ) ,   
 i(g ) = n 
 , where  
 n are said n - times unit roots.therefore , there is a reversible matrix ( 11)qn=nijn0i , jn1,such that ( 12)qn1bnqn = diag1,n,  ,nn1dn . it is not difficult to find that i 
 n2  ( 6i 
 n1  d 
 n1  d 
 n1 
 )  d 
 n2  ( i 
 n1 + d 
 n1 
 )  d 
 n2 
   ( i 
 n1 + d 
 n1 ) is a diagonal matrix whose diagonal elements are ( 16)6n1in1in2jn2jn1in2j+n1in2j=62cos2in12cos2jn22cos2in1cos2jn2,where 0  i  n 
 1  1 and  0  j  n 
 2  1.this means that the eigenvalues of the matrix l are  = 6  2cos 
 i  2cos 
 j  2cos( 
 i   
 j ) , 0  i  n 
 1  1 , and  0  j  n 
 2  1 , where i = 2i / n 
 1 and  
 j = 2j / n 
 2.by formula ( 2 ) , the laplacian - energy - like invariant is ( 17)lelmhtn1,n2=i=0n11  j=0n2162cosi2cosj2cosij . so ( 18)limn1  limn2lelmhtn1,n2n1n2=limn1  limn21n1n2i=0n11  j=0n2162cosi2cosj2cosij=0162cos2x2cos2y2cos2xydx  dy=1420262cosx2cosy2cosxydx  dy2.3705 . 

  with the proper labelling of the vertices of the modified hexagonal lattice , its laplacian matrix is 
 ( 8)where i 
 n1 , i 
 n2 are the unit matrices and m  n is tensor product of matrices m and n. consider(9)bn2=01000001000000110000n2n2 . 
  the matrix l(mh(n 
 1 , n 
 2 ) ) can be defined as follows : ( 10)lmhtn1,n2in2cn1bn2pn1bn2tpn1t = in26in1bn1bn1tbn2in1+bn1tbn2tin1+bn1 . , g 
 } be a cyclic group of order n. obviously ,  : g 
   b 
 n 
  can express the group . the cyclic group of order n has linear values of n   
 i  ( i = 0,1 ,  , n  1 ) ,   
 i(g ) =  
 n 
 , where  
 n are said n - times unit roots . therefore , there is a reversible matrix ( 11)qn=nijn0i , jn1,such that ( 12)qn1bnqn = diag1,n,  ,nn1dn . it is not difficult to find that i 
 n2  ( 6i 
 n1  d 
 n1  d 
 n1 
 )  d 
 n2  ( i 
 n1 + d 
 n1 
 )  d 
 n2 
   ( i 
 n1 + d 
 n1 ) is a diagonal matrix whose diagonal elements are ( 16)6n1in1in2jn2jn1in2j+n1in2j=62cos2in12cos2jn22cos2in1cos2jn2,where 0  i  n 
 1  1 and  0  j  n 
 2  1 . this means that the eigenvalues of the matrix l are  = 6  2cos 
 i  2cos 
 j  2cos( 
 i   
 j ) , 0  i  n 
 1  1 , and  0  j  n 
 2  1 , where  
 i = 2i / n 
 1 and  
 j = 2j / n 
 2 . by formula ( 2 ) , the laplacian - energy - like invariant is ( 17)lelmhtn1,n2=i=0n11  j=0n2162cosi2cosj2cosij . such computations would be possible on a computer with high memory and processing speed , we used mac pro with processor 2  2.93  ghz 6-core intel xeon ( 24 hyperthreads in total ) and memory 24  gb 1333  mhz ddr3 to obtain the results . 
  the numerical integration value in last line is calculated with the software matlab . as such computations would be possible on a computer with high memory and processing speed , we used mac pro with processor 2  2.93  ghz 6-core intel xeon ( 24 hyperthreads in total ) and memory 24  gb 1333  mhz ddr3 to obtain the results . by theorems 4 and 5 theorem 7 . for the modified hexagonal lattices mh 
 ( n 
 1 , n 
 2 ) , mh 
 ( n 
 1 , n 
 2 ) , and mh 
 ( n 
 1 , n 
 2 ) with toroidal , cylindrical , and free boundary conditions , then,(19)1  limn1  limn2lelmhtn1,n2n1n2=limn1  limn2lelmhcn1,n2n1n2=limn1  limn2lelmhfn1,n2n1n22.3705,2  lelmhtn1,n2=lelmhcn1,n2=lelmhfn1,n22.3705n1n2 . 

  for the modified hexagonal lattices mh 
 ( n 
 1 , n 
 2 ) , mh 
 ( n 
 1 , n 
 2 ) , and mh 
 ( n 
 1 , n 
 2 ) with toroidal , cylindrical , and free boundary conditions , then,(19)1  limn1  limn2lelmhtn1,n2n1n2=limn1  limn2lelmhcn1,n2n1n2=limn1  limn2lelmhfn1,n2n1n22.3705,2  lelmhtn1,n2=lelmhcn1,n2=lelmhfn1,n22.3705n1n2 . the modified union jack lattice with toroidal boundary condition is denoted by s 
 ( n 
 1 , n 
 2 ) . let  
 i = 2i / n 
 1 ;   
 j = 2j / n 
 2 . then ( 20)1  lelstn1,n2=i=0n11  j=0n2182cosi2cosj4cosicosj,2  limn1  limn2lelstn1,n2n1n2=1420282cosx2cosy4cosxcosy  dx  dy2.7586 . i = 2i / n 
 1 ;   
 j = 2j / n 
 2 . then ( 20)1  lelstn1,n2=i=0n11  j=0n2182cosi2cosj4cosicosj,2  limn1  limn2lelstn1,n2n1n2=1420282cosx2cosy4cosxcosy  dx  dy2.7586 . proofwith a proper labelling of the vertices of the modified union jack lattice , its laplacian matrix can be represented as ( 21)lstn1,n2=gu000ututgu0000utg000000gu0000utguu000utgn2n2,g=810001181000018000000810000181100018n1n1,u=110001111000011000000110000111100011n1n1 . let(23)qn=nijn0i , jn1,such that(24)qn1bnqn = diag1,n,  ,nn1dn . 
 actually , ( 25)bnt = bn1,qnt = qn1;consequently , ( 26)qn1bntqn = diag1,1,  ,nn1dn1 . it is not difficult to find that ( 28)in28in1dn1dn11dn2in1+dn1+dn11dn21in1+dn11+dn1is a diagonal matrix whose diagonal elements are ( 29)8n1in1in2jn2jn1in2j+n1in2j+n1in2j+n1in2j=82cos2in12cos2jn24cos2in1cos2jn2,where 0  i  n 
 1  1 and  0  j  n 
 2  1.this means that the eigenvalues of the matrix l are  = 8  2cos 
 i  2cos j  4cos 
 i 
  
 j , 0  i  n 
 1  1 , and  0  j  n 
 2  1 , where i = 2i / n 
 1 and  
 j = 2j / n 
 2.by formula ( 2 ) , the laplacian - energy - like invariant is ( 30)lelstn1,n2=i=0n11  j=0n2182cosi2cosj4cosicosj . so ( 31)limn1  limn2lelstn1,n2n1n2=limn1  limn21n1n2i=0n11  j=0n2182cosi2cosj4cosicosj=0182cos2x2cos2y4cos2xcos2y  dx  dy=1420282cosx2cosy4cosxcosy  dx  dy2.7586 . 

  with a proper labelling of the vertices of the modified union jack lattice let(23)qn=nijn0i , jn1,such that(24)qn1bnqn = diag1,n,  ,nn1dn . 
  actually , ( 25)bnt = bn1,qnt = qn1;consequently , ( 26)qn1bntqn = diag1,1,  ,nn1dn1 . it is not difficult to find that ( 28)in28in1dn1dn11dn2in1+dn1+dn11dn21in1+dn11+dn1is a diagonal matrix whose diagonal elements are ( 29)8n1in1in2jn2jn1in2j+n1in2j+n1in2j+n1in2j=82cos2in12cos2jn24cos2in1cos2jn2,where 0  i  n 
 1  1 and  0  j  n 
 2  1 . this means that the eigenvalues of the matrix l are  = 8  2cos 
 i  2cos 
 j  4cos 
 i 
  
 j , 0  i  n 
 1  1 , and  0  j  n 
 2  1 , where i = 2i / n 
 1 and  
 j = 2j / n 
 2 . by formula ( 2 ) , the laplacian - energy - like invariant is ( 30)lelstn1,n2=i=0n11  j=0n2182cosi2cosj4cosicosj . so ( 31)limn1  limn2lelstn1,n2n1n2=limn1  limn21n1n2i=0n11  j=0n2182cosi2cosj4cosicosj=0182cos2x2cos2y4cos2xcos2y  dx  dy=1420282cosx2cosy4cosxcosy  dx  dy2.7586 . 
  by theorems 4 and 8 , it is not difficult to arrive at the following theorem . theorem 9 . for the modified union jack lattices s 
 ( n 
 1 , n 
 2 ) , s 
 ( n 
 1 , n 
 2 ) , and s 
 ( n 
 1 , n 
 2 ) with toroidal , cylindrical , and free boundary conditions , then , ( 32)1  limn1  limn2lelstn1,n2n1n2=limn1  limn2lelscn1,n2n1n2=limn1  limn2lelsfn1,n2n1n22.7586,2  lelstn1,n2=lelscn1,n2=lelsfn1,n22.7586n1n2 . 

  for the modified union jack lattices s 
 ( n 
 1 , n 
 2 ) , s 
 ( n 
 1 , n 
 2 ) , and s 
 ( n 
 1 , n 
 2 ) with toroidal , cylindrical , and free boundary conditions , then , ( 32)1  limn1  limn2lelstn1,n2n1n2=limn1  limn2lelscn1,n2n1n2=limn1  limn2lelsfn1,n2n1n22.7586,2  lelstn1,n2=lelscn1,n2=lelsfn1,n22.7586n1n2 . 
  the honeycomb lattice with toroidal boundary condition , denoted by hc(n 
 1 , n 
 2 ) , can be constructed by starting with an m  n square lattice and adding two diagonal edges to each square . let  
 i = 2i / n 
 1 and  
 j = 2j / n 
 2 . then ( 33)1  lelhctn1,n2=i=0n11  j=0n213 + 3 + 2cosi+2cosj+2cosij+i=0n11  j=0n2133 + 2cosi+2cosj+2cosij,2  limn1  limn2lelhctn1,n22n1n2=182023 + 3 + 2cosx+2cosy+2cosxydx  dy+1820233 + 2cosx+2cosy+2cosxydx  dy1.6357 . let  
 i = 2i / n 
 1 and  
 j = 2j / n 
 2 . then ( 33)1  lelhctn1,n2=i=0n11  j=0n213 + 3 + 2cosi+2cosj+2cosij+i=0n11  j=0n2133 + 2cosi+2cosj+2cosij,2  limn1  limn2lelhctn1,n22n1n2=182023 + 3 + 2cosx+2cosy+2cosxydx  dy+1820233 + 2cosx+2cosy+2cosxydx  dy1.6357 . proofsimilarly , the laplacian matrix of the honeycomb lattice is lhctn1,n2=3im - f - ft3im , where m = n 
 1 
 n 
 2 and f is an m  m matrix . the matrix f can be written in the following form : ( 34)f = w000iiiw00000iw000000w00000iw00000iwn2n2,w=100001110000011000000100000110100011n1n1,where i represents the unit matrix of n 
 1  n 
 1 and i 
 m represents the unit matrix of m  m , respectively.based on theorem 5 , the matrix f can be written as ( 35)fin2wn1+bn2tin1=in2in1+bn1t+bn2tin1 . let ( 36)qn=nijn0i , jn1,such that ( 37)qn1bnqn = diag1,n,  ,nn1dn . 
 similarly , ( 38)bnt = bn1,qnt = qn1 ; 
 hence , ( 39)qn1bntqn = diag1,1,  ,nn1dn1 . it is not difficult to find that i 
 n2  ( i 
 n1 + d 
 n1 
 ) + d 
 n2 
   i 
 n1 is a diagonal matrix whose diagonal elements are 1 +  
 n1 
  +  
 n2 
 , so matrix l(hc(n 
 1 , n 
 2 ) ) can be reduced to the following form : ( 41)lhctn1,n2=31n1in2j1n1in2j3 . 
 by det(i l(hc(n 
 1 , n 
 2 ) ) ) = 0 , we can get ( 42)321n1in2j1n1in2j=3+n1i+n1i+n2j+n2j+n1in2j+n1in2j=3 + 2cos2in1 + 2cos2jn2 + 2coscos2in1cos2jn2 . 
 therefore , the l(hc(n 
 1 , n 
 2 ) ) characteristic eigenvalues are ( 43)=33 + 2cos2in1 + 2cos2jn2 + 2cos2in12jn2,where 0 i  n 
 1  1 and 0  j  n 
 2  1.let  
 i = 2i / n 
 1 and  
 j = 2j / n 
 2 . by formula ( 2 ) , we may obtain the laplacian - energy - like invariant : ( 44)lelhctn1,n2=i=0n11  j=0n213 + 3 + 2cosi+2cosj+2cosij+i=0n11  j=0n2133 + 2cosi+2cosj+2cosij . 
 by the definition of double integration , we arrive at ( 45)limn1  limn2lelhctn1,n22n1n2limn1  limn212n1n2i=0n11  j=0n213 + 3 + 2cosi+2cosj+2cosij+limn1  limn212n1n2i=0n11  j=0n2133 + 2cosi+2cosj+2cosij=12013 + 3 + 2cos2x+2cos2y+2cos2xydx  dy+120133 + 2cos2x+2cos2y+2cos2xydx  dy=182023 + 3 + 2cosx+2cosy+2cosxydx  dy+1820233 + 2cosx+2cosy+2cosxydx  dy1.6357 . similarly , the laplacian matrix of the honeycomb lattice is lhctn1,n2=3im - f - ft3im , where m = n 
 1 
 n 
 2 and f is an m  m matrix . the matrix f can be written in the following form : ( 34)f = w000iiiw00000iw000000w00000iw00000iwn2n2,w=100001110000011000000100000110100011n1n1,where i represents the unit matrix of n 
 1  n 
 1 and i 
 m represents the unit matrix of m  m , respectively . based on theorem 5 , the matrix f can be written as ( 35)fin2wn1+bn2tin1=in2in1+bn1t+bn2tin1 . let ( 36)qn=nijn0i , jn1,such that ( 37)qn1bnqn = diag1,n,  ,nn1dn . 
  similarly , ( 38)bnt = bn1,qnt = qn1 ; 
  hence , it is not difficult to find that i 
 n2  ( i 
 n1 + d 
 n1 
 ) + d 
 n2 
   i 
 n1 is a diagonal matrix whose diagonal elements are 1 +  
 n1 
  +  
 n2 
 , so matrix l(hc(n 
 1 , n 
 2 ) ) can be reduced to the following form : ( 41)lhctn1,n2=31n1in2j1n1in2j3 . 
  by det(i l(hc(n 
 1 , n 
 2 ) ) ) = 0 , we can get ( 42)321n1in2j1n1in2j=3+n1i+n1i+n2j+n2j+n1in2j+n1in2j=3 + 2cos2in1 + 2cos2jn2 + 2coscos2in1cos2jn2 . therefore , the l(hc(n 
 1 , n 
 2 ) ) characteristic eigenvalues are ( 43)=33 + 2cos2in1 + 2cos2jn2 + 2cos2in12jn2,where 0  i  n 
 1  1 and 0  j  n 
 2  1 . i = 2i / n 
 1 and  
 j = 2j / n 
 2 . by formula ( 2 ) , we may obtain the laplacian - energy - like invariant : ( 44)lelhctn1,n2=i=0n11  j=0n213 + 3 + 2cosi+2cosj+2cosij+i=0n11  j=0n2133 + 2cosi+2cosj+2cosij . 
  by the definition of double integration , we arrive at ( 45)limn1  limn2lelhctn1,n22n1n2limn1  limn212n1n2i=0n11  j=0n213 + 3 + 2cosi+2cosj+2cosij+limn1  limn212n1n2i=0n11  j=0n2133 + 2cosi+2cosj+2cosij=12013 + 3 + 2cos2x+2cos2y+2cos2xydx  dy+120133 + 2cos2x+2cos2y+2cos2xydx  dy=182023 + 3 + 2cosx+2cosy+2cosxydx  dy+1820233 + 2cosx+2cosy+2cosxydx  dy1.6357 . 
  by theorems 4 and 10 , hc(n 
 1 , n 
 2 ) , hc(n 
 1 , n 
 2 ) , and hc(n 
 1 , n 
 2 ) with toroidal , cylindrical , and free boundary conditions , then , ( 46)1  limn1  limn2lelhctn1,n2n1n2=limn1  limn2lelhccn1,n2n1n2=limn1  limn2lelhcfn1,n2n1n21.6357;2  lelhctn1,n2=lelhccn1,n2=lelhcfn1,n21.6357n1n2 . 

  for the honeycomb lattices hc(n 
 1 , n 
 2 ) , hc(n 
 1 , n 
 2 ) , and hc(n 
 1 , n 
 2 ) with toroidal , cylindrical , and free boundary conditions , then , ( 46)1  limn1  limn2lelhctn1,n2n1n2=limn1  limn2lelhccn1,n2n1n2=limn1  limn2lelhcfn1,n2n1n21.6357;2  lelhctn1,n2=lelhccn1,n2=lelhcfn1,n21.6357n1n2 . in this paper , we mainly studied the laplacian - energy - like invariants of the modified hexagonal lattice , modified jack lattice , and honeycomb lattice . the proposed results imply that the asymptotic laplacian - energy - like invariants of those lattices are independent of the three boundary conditions . the problems on the various topological indices of lattices have much important significance in the mathematical theory , chemical energy , statistical physics , and networks science .
this paper mainly studies the laplacian - energy - like invariants of the modified hexagonal lattice , modified union jack lattice , and honeycomb lattice . by utilizing the tensor product of matrices and the diagonalization of block circulant matrices 
 , we derive closed - form formulas expressing the laplacian - energy - like invariants of these lattices . in addition 
 , we obtain explicit asymptotic values of these invariants with software - aided computations of some integrals .