Analyzing lattice networks through substructures

Applied Mathematics and Computation 329 (2018) 297–314

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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc

Analyzing lattice networks through substructures Hui Lei a, Tao Li b,∗, Yuede Ma c, Hua Wang d,e,∗∗ a

Center for Combinatorics and LPMC Nankai University, Tianjin 300071, PR China College of Computer and Control Engineering Nankai University, Tianjin 300071,PR China School of Science Xi’an Technological University, Shaanxi Xi’an, 710021, PR China d College of Software Nankai University, Tianjin 300071, PR China e Department of Mathematical Sciences Georgia Southern University, Statesboro GA 30460-8093, USA b c

a r t i c l e

i n f o

Keywords: Lattice networks Graph invariants Subgraphs

a b s t r a c t Analyzing the topology of network structures is an important topic studied from many different aspects of science and mathematics. The Wiener polarity index (number of unordered pairs of vertices at distance 3 from each other) is one of the representative descriptors of graph structures. It was computed for several lattice networks by Chen et al. [11] in an effort to understand the properties of these networks. The Wiener polarity index is a variation of the classic distance-based graph invariant, the Wiener index (sum of distances between all pairs of vertices), which is known to be closely related to the number of substructures. In this paper we examine the numbers of various subgraphs of order 4 for these lattice networks. In addition to confirming their symmetric nature, comparing the numbers of various substructures leads to insights on other less trivial characteristics of these network structures of common interest. © 2018 Elsevier Inc. All rights reserved.

1. Introduction In modern graph theoretical studies an important topic is to understand a graph structure through the so-called graph invariants or topological indices. Following standard notations and terminologies in [4], we generally let G denote a graph with vertex set V(G) and edge set E(G). The distance dG (u, v ) (or simply d (u, v )) between two vertices u and v of G is the length of the shortest path that connects u and v. One of the most well-known and well-studied distance-based graph invariants [1,2] is called the Wiener number W(G), also termed as the Wiener index in the literature. It is defined as the sum of distances over all unordered vertex pairs in G [42]:

W (G ) =



dG (u, v ).

{u,v}⊆V (G )

As a classic distance-based graph invariant, properties of the Wiener index have been studied in numerous articles. Some recent papers include [17,23,26]. A variation of W(G) was also introduced by Wiener [42], called the Wiener polarity index. Denoted by Wp (G), we have

Wp (G ) = |{(u, v )|dG (u, v ) = 3, u, v ∈ V (G )}|. ∗ ∗∗

Corresponding author. Corresponding author at: Department of Mathematical Sciences Georgia Southern University, Statesboro GA 30460-8093, USA. E-mail addresses: [email protected] (H. Lei), [email protected] (T. Li), mayuede0 0 0 [email protected] (Y. Ma), [email protected] (H. Wang).

https://doi.org/10.1016/j.amc.2018.02.012 0 096-30 03/© 2018 Elsevier Inc. All rights reserved.

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For applications and theoretical studies on Wiener index and Wiener polarity index, we refer to [16,19,22,25–27,37,38,46,47]. Similar work has been done on many other graph invariants, such as degree-based indices [3,30], spectrum-based indices [11–13,20,21,24,31,36,44,45], entropy indices [8,9,14,29] and others [5–7,34]. A lattice graph, or simply a lattice, is a graph that can be embedded in a Euclidean space Rn to form a regular tiling. It is not difficult to notice the symmetric nature of such structures. For exactly this reason, lattice networks appear to be among the most common network structures [15,18]. Consequently, various graph invariants have been studied for different lattice networks [28,32,33,35,39,43]. In particular, Chen et al. [10] computed the Wiener polarity index of the square lattices, the hexagonal lattices, the triangular lattices, and the 33 · 42 lattices, each corresponding to a grid with specific geometric shapes. There is a well known correlation between the distance-based graph invariants (the Wiener index in particular) and the numbers of substructures (see for instance [41] and the survey [40]), that graphs with smaller sum of distances usually have more substructures. Consequently it is natural to study the number of substructures of a graph structure after the distance-based graph invariants are examined. On the other hand, note that the Wiener polarity index of a graph is simply the number of unordered pairs of vertices at distance 3, it makes sense to consider the number of paths of length 3 or more generally the number of subgraphs of order 4. For this purpose, with a given collection S of graphs we let FS (G) denote the number of subgraphs of G that are isomorphic to a graph in S. If S = {A}, we simply write FA (G) instead of FS (G). We will first consider the case S = {P4 } and compute FS (G) for the aforementioned lattice networks in Sections 2–5. We postpone the definitions of each lattice network to these individual sections. As a quick comparison with [10], it is easy to see that when m, n → ∞, we have

⎧ ⎪ ⎨3,

if if if if

FP4 (G ) 4/3, ∼ Wp ( G ) ⎪ ⎩69/9, 74/15, FP (G ) 4

The large value of

W p (G )

G G G G

is is is is

the the the the

square lattice; hexagonal lattice; triangular lattice; 33 · 42 lattice.

(2)

for the triangular lattice and the 33 · 42 lattice immediately imply that there are generally more

edges in these lattices than the first two, which resulted in the fact that there are dramatically more paths of length 3 than pairs of vertices at distance 3. With this in mind we should also be expecting more substructures in the latter two lattices than the first two, which we will see in Section 6 when we consider FS (G) for other substructures of order 4. In Section 7 we summarize our study with observations on the characteristics of these lattice networks (including their boundary conditions). Before moving on to specific lattice networks we recall some basic notations and present a simple general formulation for FP4 (G ) of a graph G. • Given sets A and B, the Cartesian product of A and B, generally denoted by AB, is the set of all ordered pairs (a, b) with a ∈ A and b ∈ B. That is,

AB = {(a, b)|a ∈ A, b ∈ B}. Given graphs G and H with vertex sets U and V, the Cartesian product GH is a graph with vertex set UV, and any two vertices (u, u ) and (v, v ) are adjacent in GH if and only if either u = v and u is adjacent to v in H, or u = v and u is adjacent to v in G. It is easy to visualize that the concept of Cartesian product of graphs will play an important role in the study of lattice networks. • For a graph G and vertex v ∈ V (G ), let PG4 (v ) be the set of length-3 paths (in G) with v as one of the endpoints. By definition we have



FP4 (G ) =

v∈V (G )

|PG4 (v )| 2

.

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This formulation of FP4 (G ), although trivial, will facilitate our study in later sections. 2. The number of length-3 paths of square lattices First note that with Cartesian product of graphs, we have Pm Pn (m ≥ 2, n ≥ 2), Pm Cn (m ≥ 2, n ≥ 3), and Cm Cn (m ≥ 3, n ≥ 3) as exactly the square lattices with free, cylindrical and toroidal boundary conditions, respectively (see Fig. 1). Without loss of generality, we assume that n ≥ m in Pm Pn and Cm Cn in our main result. Theorem 1. Let Pm Pn , Pm Cn and Cm Cn denote the square lattices with free, cylindrical and toroidal boundary conditions, respectively. Then we have

FP4 (Pm Pn ) =

⎧ ⎨4,

i f m = 2, n = 2;

12n − 22,

i f m = 2, n ≥ 3;

18mn − 25m − 25n + 28,

i f m ≥ 3, n ≥ 3;

⎩

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H. Lei et al. / Applied Mathematics and Computation 329 (2018) 297–314

299

am am−1 ··· ··· a2 a1

b1

b2

· · · · · · bn−1 bn

Fig. 1. The square lattice.

⎧ ⎪ ⎨30,

if if if if

12n, FP4 (Pm Cn ) = ⎪ ⎩51m − 75, 18mn − 25n,

FP4 (Cm Cn ) =

⎧ ⎨144,

m = 2, n = 3; m = 2, n ≥ 4; m ≥ 3, n = 3; m ≥ 3, n ≥ 4;

i f m = 3, n = 3;

51n,

i f m = 3, n ≥ 4;

18mn,

i f m ≥ 4, n ≥ 4.

⎩

(5)

(6)

Proof. (i) First, under the assumption that n ≥ m, we consider the square lattice Pm Pn with free boundary condition. After examining the initial cases we compute FP4 (Pm Pn ) through the recursion

FP4 (Pm Pn ) =

m 

(|PP4m Pn (ai )| ) − FP4 (Pm ) + FP4 (Pm Pn−1 ).

i=1

Here the term

m  i=1

(|PP4m Pn (ai )| ) − FP4 (Pm ) counts the length-3 paths in Pm Pn but not in Pm Pn−1 .

Case 1 When m = 2: • It is easy to see that FP4 (P2 P2 ) = 4 and FP4 (P2 P3 ) = 14. • For n ≥ 4, we have |PP4 Pn (a1 )| = |PP4 Pn (a2 )| = 6. Thus 2

FP4 (P2 Pn ) =

2 

2

(|PP42 Pn (ai )| ) − FP4 (P2 ) + FP4 (P2 Pn−1 )

i=1

= 12 + FP4 (P2 Pn−1 ) = ··· = 12(n − 3 ) + FP4 (P2 P3 ) = 12n − 22. Note that FP4 (P2 P3 ) = 14 also satisfies this general formula. Case 2 When m = 3: • If n = 3, FP4 (P3 P3 ) = 40. • If n ≥ 4, we have |PP4 Pn (a1 )| = |PP4 Pn (a3 )| = 9 and |PP4 Pn (a2 )| = 11. Thus 3

FP4 (P3 Pn ) =

3 

3

3

(|PP43 Pn (ai )| ) − FP4 (P3 ) + FP4 (P3 Pn−1 )

i=1

= 29 + FP4 (P3 Pn−1 ) = ··· = 29(n − 3 ) + FP4 (P3 P3 ) = 29n − 47. This is also satisfied by FP4 (P3 P3 ) = 40.

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Case 3 When m = 4, for n ≥ m we have |PP4 Pn (a1 )| = |PP4 Pn (a4 )| = 10 and |PP4 Pn (a2 )| = |PP4 Pn (a3 )| = 14. Hence 4

FP4 (P4 Pn ) =

4 

4

4

4

(|PP44 Pn (ai )| ) − FP4 (P4 ) + FP4 (P4 Pn−1 )

i=1

= 47 + FP4 (P4 Pn−1 ) = ··· = 47(n − 3 ) + FP4 (P4 P3 ) = 47(n − 3 ) + FP4 (P3 P4 ) = 47n − 72. Case 4 When m = 5, similar to Case 3 we have

FP4 (P5 Pn ) = 65n − 97 for n ≥ m = 5. Case 5 In general, when m ≥ 6 and n ≥ m. Following similar computation we have

|PP4m Pn (a1 )| = |PP4m Pn (am )| = 10; |PP4m Pn (a2 )| = |PP4m Pn (am−1 )| = 15; |PP4m Pn (a3 )| = |PP4m Pn (am−2 )| = 18; |PP4m Pn (ai )| = 19 for 4 ≤ i ≤ m − 3. Since FP4 (P3 Pm ) = FP4 (Pm P3 ) = 29m − 47,

FP4 (Pm Pn ) =

m 

(|PP4m Pn (ai )| ) − FP4 (Pm ) + FP4 (Pm Pn−1 )

i=1

= 18m − 25 + FP4 (Pm Pn−1 ) = ··· = (18m − 25 ) × (n − 3 ) + FP4 (Pm P3 ) = (18m − 25 ) × (n − 3 ) + FP4 (P3 Pm ) = 18mn − 25m − 25n + 28. Lastly we note that this formula coincides with our findings for smaller values of m. Hence FP4 (Pm Pn ) = 18mn − 25m − 25n + 28 for n ≥ 3 and m ≥ 3. (ii) In the square lattice Pm Cn with cylindrical boundary condition, the vertices from the same row are endpoints of the same number of length-3 paths. Thus we only need to compute PP4m Cn (ai ) for each i = 1, 2, . . . , m. Case 1 When n ≥ 4: • If m = 2, we have |PP4 Cn (a1 )| = |PP4 Cn (a2 )| = 12 and consequently 2

2

12 × 2n FP4 (P2 Cn ) = = 12n. 2 • If m = 3, we have |PP4 Cn (a1 )| = |PP4 Cn (a3 )| = 18 and |PP4 Cn (a2 )| = 22. Thus 3

3

3

18 × 2n + 22n FP4 (P3 Cn ) = = 29n. 2 • If m = 4, we have |PP4 Cn (a1 )| = |PP4 Cn (a4 )| = 19 and |PP4 Cn (a2 )| = |PP4 Cn (a3 )| = 28. Thus 4

4

4

4

19 × 2n + 28 × 2n FP4 (P4 Cn ) = = 47n. 2 • If m = 5, we have |PP4 Cn (a1 )| = |PP4 Cn (a5 )| = 19, |PP4 Cn (a2 )| = |PP4 Cn (a4 )| = 29, and |PP4 Cn (a3 )| = 34. Thus 5

5

19 × 2n + 29 × 2n + 34n FP4 (P5 Cn ) = = 65n. 2 • In general, for m ≥ 6, we have

|PP4m Cn (a1 )| = |PP4m Cn (am )| = 19; |PP4m Cn (a2 )| = |PP4m Cn (am−1 )| = 29; |PP4m Cn (a3 )| = |PP4m Cn (am−2 )| = 35;

5

5

5

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301

Fig. 2. The hexagonal lattice.

|PP4m Cn (ai )| = 36, 4 ≤ i ≤ m − 3. Thus

19 × 2n + 29 × 2n + 35 × 2n + 36 × (m − 6 )n 2 = 18mn − 25n.

FP4 (Pm Cn ) =

It is easy to check that this general formula also holds for m = 3, 4, 5. Case 2 When n = 3: |PP4m C (ai )| = |PP4m Cn (ai )| − 2 for 1 ≤ i ≤ m, n ≥ 4. Hence FP4 (P2 C3 ) = 12 × 3 − 3 × 2 = 30 and 3

FP4 (Pm C3 ) = 18m × 3 − 25 × 3 − 3m = 51m − 75. (iii) Lastly, we consider Cm Cn with n ≥ m. In this case every vertex is the end point of the same number of length-3 paths. For m ≥ 4, we have |PC4m Cn (a1 )| = 36. Thus

FP4 (Cm Cn ) =

36mn = 18mn. 2

For m = 3: • If n = 3, we have |PC4 C (a1 )| = 32. Thus 3

3

32mn FP4 (C3 C3 ) = = 144. 2 • If n ≥ 4, we have |PC4 Cn (a1 )| = 34. Thus 3

FP4 (C3 Cn ) =

34mn = 51n. 2

 3. The number of length-3 paths of the hexagonal lattices Following the notations in [43], the hexagonal lattices with toroidal, cylindrical and free boundary conditions are denoted by Ht (n, m), Hc (n, m) and Hf (n, m), respectively, where (a1 , b1 ), (a2 , b2 ), . . . , (am+1 , bm+1 ); (a1 , c1∗ ), (c1 , c2∗ ), (c2 , c3∗ ), . . . , (cn−1 , cn∗ ), (cn , bm+1 ) are edges in Ht (n, m) (as illustrated in Fig. 2). It is easy to see, from the definition, that |V (Ht (n, m ))| = |V (H c (n, m ))| = |V (H f (n, m ))| = 2(n + 1 )(m + 1 ). Theorem 2. For the hexagonal lattices Ht (n, m), Hc (n, m) and Hf (n, m) with toroidal, cylindrical and free boundary conditions, we have

FP4 (H t (n, m )) = 12(n + 1 )(m + 1 ), FP4 (H c (n, m )) = 12m(n + 1 ),

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and

FP4 (H f (n, m )) = 12nm − 2. Proof. (i) In Ht (n, m), first note the simple fact that |PH4t (n,m ) (· )| is the same for every vertex in Ht (n, m). Take vertex a1 from Fig. 2 for instance, for n ≥ 1 and m ≥ 1, we have

|PH4t (n,m) (a1 )| = 12. Hence Eq. (3) implies that

FP4 (H t (n, m )) =

12 × 2(n + 1 )(m + 1 ) = 12(n + 1 )(m + 1 ). 2

(ii) In the case of Hc (n, m) (Fig. 2), we first partition the vertex set of Hc (n, m) into m + 1 disjoint sets such that vertices in the same set share the same value for |PH4t (n,m ) (· )|: •

V0 = {a1 , c1 , c2 , . . . , cn , c1∗ , c2∗ , . . . , cn∗ , bm+1 }, •

V1 = {u11 , u12 , . . . , u1n , b1 , am+1 , vm1 , vm2 , . . . , vmn }, •

V2 = {a2 , v11 , v12 , . . . , v1n , um1 , um2 , . . . , umn , bm }, .. . •

Vm = {u( m+1 )1 , . . . , u( m+1 )n , b m+1 , a( m+1 )+1 , v( m+1 )1 , . . . , v( m+1 )n } 2

2

2

2

2

2

when m is odd, and

Vm = {a m2 +1 , v m2 1 , v m2 2 , . . . , v m2 n , u( m2 +1)1 , u( m2 +1)2 , . . . , u( m2 +1)n , b m2 +1 } when m is even. We only need to compute |PH4t (n,m ) (· )| for one vertex from each class. • If m = 1, we have |PH4c (n,1 ) (a1 )| = |PH4c (n,1 ) (u11 )| = 6. Thus

FP4 (H c (n, 1 )) =

6 × 4 (n + 1 ) = 12(n + 1 ). 2

• If m = 2, we have |PH4c (n,2 ) (a1 )| = 6, |PH4c (n,2 ) (u11 )| = 8, and |PH4c (n,2 ) (a2 )| = 10. Thus

FP4 (H c (n, 2 )) =

(6 + 8 + 10 ) × 2(n + 1 ) 2

= 24(n + 1 ).

• If m ≥ 3, we have |PH4c (n,m ) (a1 )| = 6, |PH4c (n,m ) (u11 )| = 8, |PH4c (n,m ) (a2 )| = 10, |PH4c (n,m ) (ai )| = 12 for 3 ≤ i ≤

|PH4c (n,m) (ui1 )| = 12 for 2 ≤ i ≤



FP4 (H c (n, m )) =

m 2

v∈V (H c (n,m ))

. Let V  = V (H c (n, m )) − V0 − V1 − V2 , by Eq. (3) we have

m 2

+ 1, and

|PH4c (n,m) (v )|

2 6 × |V0 | + 8 × |V1 | + 10 × |V2 | + 12 × |V  | = 2 = 12m(n + 1 ).

It is not hard to check that FP4 (H c (n, 1 )) (with m = 1) and FP4 (H c (n, 2 )) (with m = 2) also satisfy this expression. (iii) Lastly, we will evaluate the number of length-3 paths of Hf (n, m) recursively. First consider the case n = 1. Without loss of generality, we can assume m ≥ n. It is easy to see that FP4 (H f (1, 1 )) = 10. For m ≥ 2, we have the following:

|PH4 f (1,m) (a1 )| = |PH4 f (1,m) (bm+1 )| = 2; |PH4 f (1,m) (u11 )| = |PH4 f (1,m) (vm1 )| = 3; |PH4 f (1,m) (a2 )| = |PH4 f (1,m) (bm )| = 4;

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303

|PH4 f (1,m) (ui1 )| = |PH4 f (1,m) (v(m+1−i)1 )| = 6 f or 2 ≤ i ≤ m − 1; |PH4 f (1,m) (ai )| = |PH4 f (1,m) (bm+2−i )| = 6 f or 3 ≤ i ≤ m; |PH4 f (1,m) (um1 )| = |PH4 f (1,m) (v11 )| = 5; |PH4 f (1,m) (am+1 )| = |PH4 f (1,m) (b1 )| = 5; |PH4 f (1,m) (c1∗ )| = |PH4 f (1,m) (c1 )| = 3. Thus



FP4 (H f (1, m )) =

v∈V (H f (1,m ))

|PH4 f (1,m) (v )|

2 [2 + 3 + 4 + 6(m − 2 ) + 6(m − 2 ) + 5 + 5 + 3] × 2 = 2 = 12m − 2,

also satisfied by FP4 (H f (1, 1 )) = 10. Now let n ≥ 2. Simple computation yields

|PH4 f (n,m) (a1 )| = 2; |PH4 f (n,m) (u11 )| = 4; |PH4 f (n,m) (a2 )| = 4; |PH4 f (n,m) (ui1 )| = 8 f or 2 ≤ i ≤ m − 1; |PH4 f (n,m) (ai )| = 6 f or 3 ≤ i ≤ m; |PH4 f (n,m) (um1 )| = 7; |PH4 f (n,m) (am+1 )| = 5; |PH4 f (n,m) (c1∗ )| = 5. Together with the structure of Hf (n, m) we now have

FP4 (H f (n, m )) =

m +1  i=1

|PH4 f (n,m) (ai )| +

m  i=1

|PH4 f (n,m) (ui1 )| + |PH4 f (n,m) c1∗ |

−FP4 (P2m+2 ) + FP4 (H f (n − 1, m )) = 12m + FP4 (H f (n − 1, m )) = 12m + 12m + FP4 (H f (n − 2, m )) = ······ = 12m × (n − 1 ) + FP4 (H f (1, m )) = 12nm − 2. Here FP4 (P2m+2 ) appears because of the double counted terms. Again this formula can be verified for n = 1. Thus we conclude that FP4 (H f (n, m )) = 12nm − 2 for n ≥ 1 and m ≥ 1.  4. The number of length-3 paths of the triangular lattices The triangular lattices with toroidal, cylindrical and free boundary conditions are respectively denoted by Tt (n, m), Tc (n, m) and Tf (n, m). It is not hard to see, that the triangular lattice with toroidal boundary condition Tt (n, m) can be considered as an n × m square lattice Cn Cm with toroidal boundary condition with an additional diagonal edge added to every square. As in Fig. 3, (a1 , a∗1 ), (a2 , a∗2 ), . . . , (am , a∗m ); (b1 , b∗1 ), (b2 , b∗2 ), . . . , (bn , b∗n ); (b2 , b∗1 ), (b3 , b∗2 ), . . . , (bn , b∗n−1 ), (b1 , b∗n ) = (a1 , a∗m ); (a2 , a∗1 ), (a3 , a∗2 ), . . . , (am , a∗m−1 ) are edges with a1 = b1 , a∗1 = bn , am = b∗1 and a∗m = b∗n . In the rest of this section we assume n ≥ 3 and m ≥ 3 for all of the triangular lattices under consideration. Theorem 3. Let Tf (n, m), Tc (n, m) and Tt (n, m) be the triangular lattices with free, cylindrical and toroidal boundary conditions, respectively. Then we have (i) For n ≥ 3 and m ≥ 3, FP4 (T f (n, m )) = 69nm − 116n − 116m + 185; (ii)



FP4 (T c (n, m )) =

204m − 348 if n = 3, m ≥ 3; 69nm − 116n if n ≥ 4, m ≥ 3;

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Fig. 3. The triangular lattice.

(iii)

FP4 (T t (n, m )) =

603 if n = 3, m = 3; 204n if n ≥ 4, m = 3; 69nm if n ≥ 4, m ≥ 4.

Proof. (i) First consider Tf (n, m) with n ≥ m. We will make use of the recursion

FP4 (T f (n, m )) =

m  i=1

Here the term

m  i=1

(|PT4f (n,m) (ai )| ) + 3m − 2 + FP4 (T f (n − 1, m )).

(|PT4f (n,m) (ai )| ) + 3m − 2 counts the length-3 paths in Tf (n, m) but not in T f (n − 1, m ).

Case 1 When m = 3: • If n = 3, FP4 (T f (3, 3 )) = 110. • If n ≥ 4, we have |P 4f (a1 )| = 35, |P4f T (n,m )

T (n,m )

|PT4f (n,m) (a3 )| = 19. Thus FP4 (T f (n, 3 )) =

3  i=1

(a2 )| = 30 and

|PT4f (n,m) (ai )| + 7 + FP4 (T f (n − 1, 3 ))

= 91 + FP4 (T f (n − 1, 3 )) = 91 +

3  i=1

|PT4f (n,m) (ai )| + 7 + FP4 (T f (n − 2, 3 ))

= 91 + 91 + FP4 (T f (n − 2, 3 )) = ··· = 91(n − 3 ) + FP4 (T f (3, 3 )) = 91n − 163. Case 2 When m = 4, for n ≥ m we have

|PT4f (n,m) (a1 )| = 43; |PT4f (n,m) (a2 )| = 50; |PT4f (n,m) (a3 )| = 37; |PT4f (n,m) (a4 )| = 20. Hence

FP4 (T f (n, 4 )) =

4  i=1

|PT4f (n,m) (ai )| + 10 + FP4 (T f (n − 1, 4 ))

= 160 + FP4 (T f (n − 1, 4 ))

H. Lei et al. / Applied Mathematics and Computation 329 (2018) 297–314

= 160 +

4  i=1

305

|PT4f (n,m) (ai )| + 10 + FP4 (T f (n − 2, 4 ))

= 160 + 160 + FP4 (T f (n − 2, 4 )) = ··· = 160(n − 3 ) + FP4 (T f (3, 4 )) = 160(n − 3 ) + FP4 (T f (4, 3 )) = 160n − 279. Case 3 When m = 5, similar to Case 2 we have

FP4 (T f (n, 5 )) = 229n − 395 for n ≥ m = 5. Case 4 When m ≥ 6 and n ≥ m. Simple computation yields

|PT4f (n,m) (a1 )| = 43; |PT4f (n,m) (a2 )| = 58; |PT4f (n,m) (a3 )| = 65; |PT4f (n,m) (a4 )| = · · · = |PT4f (n,m) (am−3 )| = 66; |PT4f (n,m) (am−2 )| = 58; |PT4f (n,m) (am−1 )| = 38; |PT4f (n,m) (am )| = 20. Also note that FP4 (T f (3, m )) = FP4 (T f (m, 3 )) = 91m − 163, we have

FP4 (T f (n, m )) =

m  i=1

|PT4f (n,m) (ai )| + 3m − 2 + FP4 (T f (n − 1, m ))

= (69m − 116 ) + FP4 (T f (n − 1, m )) = (69m − 116 ) +

m  i=1

|PT4f (n,m) (ai )| + 3m − 2

+FP4 (T f (n − 2, m )) = (69m − 116 ) + (69m − 116 ) + FP4 (T f (n − 2, m )) = ··· = (69m − 116 ) × (n − 3 ) + FP4 (T f (3, m )) = 69nm − 116n − 116m + 185. After checking this formula for smaller values of m as presented above, we conclude that FP4 (T f (n, m )) = 69nm − 116n − 116m + 185 for n ≥ 3 and m ≥ 3. (ii) Similar to before, vertices from the same row in Tc (n, m) have the same value for |PT4c (n,m ) (· )|. Thus for FP4 (T c (n, m )) it suffices to compute |PT4c (n,m ) (ai )| for i = 1, 2, . . . , m. Case 1 When n ≥ 4:

• If m = 3, we have |PT4c (n,3 ) (a1 )| = |PT4c (n,3 ) (a3 )| = 58 and |PT4c (n,3 ) (a2 )| = 66. Thus

FP4 (T c (n, 3 )) =

58 × 2n + 66n = 91n. 2

• If m = 4, we have |PT4c (n,4 ) (a1 )| = |PT4c (n,4 ) (a4 )| = 66 and |PT4c (n,4 ) (a2 )| = |PT4c (n,4 ) (a3 )| = 94. Thus

FP4 (T c (n, 4 )) =

66 × 2n + 94 × 2n = 160n. 2

• If m = 5, we have |PT4c (n,5 ) (a1 )| = |PT4c (n,5 ) (a5 )| = 66, |PT4c (n,5 ) (a2 )| = |PT4c (n,5 ) (a4 )| = 102, and |PT4c (n,5 ) (a3 )| = 122. Thus

FP4 (T c (n, 5 )) =

66 × 2n + 102 × 2n + 122n = 229n. 2

• If m ≥ 6, we have

|PT4c (n,m) (a1 )| = |PT4c (n,m) (am )| = 66; |PT4c (n,m) (a2 )| = |PT4c (n,m) (am−1 )| = 102;

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Fig. 4. The 33 · 42 lattice.

|PT4c (n,m) (a3 )| = |PT4c (n,m) (am−2 )| = 130; |PT4c (n,m) (ai )| = 138, 4 ≤ i ≤ m − 3. Thus

66 × 2n + 102 × 2n + 130 × 2n + 138 × (m − 6 )n 2 = 69nm − 116n.

FP4 (T c (n, m )) =

Again this formula can be verified for m = 3, 4, 5, consequently

FP4 (T c (n, m )) = 69nm − 116n for n ≥ 4 and m ≥ 3. Case 2 When n = 3: |PT4c (3,m ) (ai )| = |PT4c (n,m ) (ai )| − 2 for 1 ≤ i ≤ m, n ≥ 4. Hence

FP4 (T c (3, m )) = 69m × 3 − 116 × 3 − 3m = 204m − 348. (iii) Now for Tt (n, m) with n ≥ m, the value of |PT4t (n,m ) (· )| is the same for all vertices. Suppose m ≥ 4. We have |PT4t (n,m ) (a1 )| = 138 and hence

FP4 (T t (n, m )) =

138mn = 69mn. 2

Suppose m = 3: • If n = 3, we have |PT4t (3,3 ) (a1 )| = 134 and hence FP4 (T t (3, 3 )) = 603.

• If n ≥ 4, we have |PT4t (n,3 ) (a1 )| = 136 and hence FP4 (T t (n, 3 )) = 204n.  5. The number of length-3 paths of the 33 · 42 lattices The 33 · 42 lattice with toroidal boundary condition, denoted by St (n, 2m), can be constructed from the square lattice C2m Cn by adding a diagonal edge in each square of every other row, as shown in Fig. 4. Here a1 = b1 , a2m = b∗1 , a∗1 = bn , a∗2m = b∗n , and (a1 , a∗1 ), (a2 , a∗2 ), . . . , (a2m , a∗2m ); (b1 , b∗1 ), (b2 , b∗2 ), . . . , (bn , b∗n ); (a1 , a∗2 ), (a3 , a∗4 ), . . . , (a2m−1 , a∗2m ) are edges. For the rest of this section we will assume n ≥ 3 and m ≥ 2 when the 33 · 42 lattices are discussed. Theorem 4. Let Sf (n, 2m), Sc (n, 2m) and St (n, 2m) be the 33 · 42 lattices with free, cylindrical and toroidal boundary conditions, respectively. Then (i) For n ≥ 3 and m ≥ 2, FP4 (S f (n, 2m )) = 74nm − 46n − 114m + 56;

H. Lei et al. / Applied Mathematics and Computation 329 (2018) 297–314

(ii)

307



216m − 138 if n = 3, m ≥ 2; 74nm − 46n if n ≥ 4, m ≥ 2;

FP4 (S (n, 2m )) = c

(iii)



216m if n = 3, m ≥ 2; 74nm if n ≥ 4, m ≥ 2.

FP4 (S (n, 2m )) = t

Proof. (i) For Sf (n, 2m) with free boundary condition, first we consider FP4 (S f (3, 2m )). Since |P 4f

|PS4f (3,2m) (a∗2m+1−i )| for 1 ≤ i ≤ 2m, for m ≥ 4 we have the following:

S ( 3,2m )

( ai )| =

|PS4f (3,2m) (a1 )| = |PS4f (3,2m) (a∗2m )| = 16; |PS4f (3,2m) (a2 )| = |PS4f (3,2m) (a∗2m−1 )| = 24; |PS4f (3,2m) (a3 )| = |PS4f (3,2m) (a∗2m−2 )| = 30; |PS4f (3,2m) (a2m−2 )| = |PS4f (3,2m) (a∗3 )| = 35; |PS4f (3,2m) (a2m−1 )| = |PS4f (3,2m) (a∗2 )| = 27; |PS4f (3,2m) (a2m )| = |PS4f (3,2m) (a∗1 )| = 23; |PS4f (3,2m) (a2i )| = |PS4f (3,2m) (a∗2m+1−2i )| = 36 for 2 ≤ i ≤ m − 2; |PS4f (3,2m) (a2i+1 )| = |PS4f (3,2m) (a∗2m−2i )| = 32 for 2 ≤ i ≤ m − 2; |PS4f (3,2m) (b2 )| = |PS4f (3,2m) (b∗2 )| = 21; |PS4f (3,2m) (c2 )| = |PS4f (3,2m) (c2m−1 )| = 28; |PS4f (3,2m) (c3 )| = |PS4f (3,2m) (c2m−2 )| = 38; |PS4f (3,2m) (ci )| = 40 for 4 ≤ i ≤ 2m − 3. Plugging into Eq. (3), we have FP4 (S f (3, 2m )) = 108m − 82 for m ≥ 4. Since FP4 (S f (3, 4 )) = 134 when m = 2 and FP4 (S f (3, 6 )) = 242 when m = 3, we conclude that FP4 (S f (3, 2m )) = 108m − 82 for m ≥ 2. Now suppose n ≥ 4. Observe that if the vertices in the first column are removed from Sf (n, 2m), then S f (n − 1, 2m ) is obtained. For the vertices in the first column of Sf (n, 2m) we have :

|PS4f (n,2m) (a1 )| = 17; |PS4f (n,2m) (a2 )| = 28; |PS4f (n,2m) (a3 )| = 31; |PS4f (n,2m) (ai )| = 40 for i = 4, 6, . . . , 2m − 4; |PS4f (n,2m) (ai )| = 33 for i = 5, 7, . . . , 2m − 3; |PS4f (n,2m) (a2m−2 )| = 39; |PS4f (n,2m) (a2m−1 )| = 28; |PS4f (n,2m) (a2m )| = 27. Thus, for m ≥ 4, we have

FP4 (S f (n, 2m )) =

2m  i=1

|PS4f (n,2m) (ai )| + m + 3 + FP4 (S f (n − 1, 2m ))

= (74m − 46 ) + FP4 (S f (n − 1, 2m ))

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H. Lei et al. / Applied Mathematics and Computation 329 (2018) 297–314

= ··· = (74m − 46 ) × (n − 3 ) + FP4 (S f (3, 2m )) = 74nm − 46n − 114m + 56. Here

2 m  i=1

|PS4f (n,2m) (ai )| + m + 3 enumerates the number of lost length-3 paths from Sf (n, 2m) to S f (n − 1, 2m ) . We skip the

details. Again this formula can be verified for small values of n and m. Hence FP4 (S f (n, 2m )) = 74nm − 46n − 114m + 56 for m ≥ 2. (ii) In the case of Sc (n, 2m) vertices in the same row share the same value for |PS4c (n,2m ) (· )|. Hence it is sufficient to compute |PS4c (n,2m ) (ai )| for i = 1, 2, . . . , 2m. Case 1 When n ≥ 4;

• If m = 2, we have |PS4c (n,4 ) (a1 )| = |PS4c (n,4 ) (a4 )| = 46 and |PS4c (n,4 ) (a2 )| = |PS4c (n,4 ) (a3 )| = 56. Thus

FP4 (Sc (n, 4 )) =

(46 + 56 ) × 2n 2

= 102n.

• If m ≥ 3, we have

|PS4c (n,2m) (a1 )| = |PS4c (n,2m) (a2m )| = 46; |PS4c (n,2m) (a2 )| = |PS4c (n,2m) (a2m−1 )| = 58; |PS4c (n,2m) (a3 )| = |PS4c (n,2m) (a2m−2 )| = 72; |PS4c (n,2m) (ai )| = 74 for 4 ≤ i ≤ 2m − 3. Then Eq. (3) implies

FP4 (Sc (n, 2m )) =

(46 + 58 + 72 ) × 2n + 74 × (2m − 6 )n 2

= 74nm − 46n. Case 2 When n = 3, |PS4c (3,m ) (ai )| = |PS4c (n,m ) (ai )| − 2 for 1 ≤ i ≤ 2m, n ≥ 4. Then

FP4 (Sc (n, 2m )) = 74m × 3 − 46 × 3 − 6m = 216m − 138. (iii) Now for St (n, 2m), as in the cases of Ht (n, m) and Tt (n, m), all vertices of St (n, 2m) are end vertices of the same number of length-3 paths. Since |V (St (n, 2m ))| = 2nm, we have the following: m For n = 3, we have |PS4t (3,2m ) (a1 )| = 72 and hence FP4 (St (3, 2m )) = 72×6 = 216m. 2 For n ≥ 4, we have |PS4t (n,2m ) (a1 )| = 74 and hence FP4 (St (n, 2m )) =

74×2nm 2

= 74nm.



6. Subgraphs of order 4 in lattice networks After enumerating length-3 paths in various lattice networks, the natural next step to further analyze these structures is to consider the same question for other subgraphs of order 4. In this section we consider K1, 3 (the star/claw on 4 vertices), C4 (the cycle on 4 vertices), C3+ (formed from appending an edge to a C3 ), and K4− (formed from removing an edge from K4 ). Let δ ij denote the Kronecker delta, i.e., the function is 1 if i = j, and 0 otherwise. 6.1. FK1,3 (G )



Suppose S = {K1,3 }. For any v ∈ V (G ) with degree d, the number of K1,3 with v as the center is d3 . Let nG (d) denote the number of vertices with degree d in G. We now list the information on vertex degrees in each of the lattice networks. (i) The square lattices: • Pm Pn :

  FK1,3 (Pm Pn ) = 2(n − 2 ) + 2(m − 2 ) + (n − 2 )(m − 2 ) = 4mn − 6n − 6m + 8.

4 3

H. Lei et al. / Applied Mathematics and Computation 329 (2018) 297–314

• Pm Cn :

  FK1,3 (Pm Cn ) = 2n + n(m − 2 )

4 3

= 4nm − 6n.

• For Cm Cn , every vertex is of degree 4. Hence

 

FK1,3 (Cm Cn ) = nm

4 3

= 4nm.

(ii) The hexagonal lattices: • Hf (n, m):

FK1,3 (H f (n, m )) = 2nm. • Hc (n, m):

FK1,3 (H c (n, m )) = 2nm + 2m. • For Ht (n, m), every vertex is of degree 3. Hence

FK1,3 (H t (n, m )) = 2(n + 1 )(m + 1 ). (iii) The triangular lattices: • Tf (n, m):

  FK1,3 (T (n, m )) = 2 + [2(n − 2 ) + 2(m − 2 )] f

  + (n − 2 )(m − 2 )

4 3

6 3

= 20nm − 32n − 32m + 50. •

Tc (n,

m):

 

4 FK1,3 (T (n, m )) = 2n 3 c

  + n (m − 2 )

6 3

• For Tt (n, m), every vertex is of degree 6. Hence

 

FK1,3 (T (n, m )) = nm t

6 3

= 20nm.

= 20nm − 32n.

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(iv) The 33 · 42 lattices: • Sf (n, 2m):

  FK1,3 (S f (n, 2m )) = 2m + [2(m − 1 ) + 2(n − 2 )]

  + (2m − 2 )(n − 2 )

4 3

5 3

= 20nm − 12n − 30m + 16. •

Sc (n,

2m):

  FK1,3 (Sc (n, 2m )) = 2n

4 3

  + n ( 2m − 2 )

5 3

= 20nm − 12n.

• For St (n, 2m), every vertex is of degree 5. Hence

 

FK1,3 (St (n, 2m )) = 2nm

5 3

= 20nm.

6.2. FC4 (G ) The numbers of 4-cycles in various lattices are rather easy to determine, we list the results below without proofs. (i) The square lattices:

FC4 (Pm Pn ) = (n − 1 )(m − 1 ). FC4 (Pm Cn ) = n(m − 1 ) + δ4n m. FC4 (Cm Cn ) = nm + δ4n m + δ4m n. (ii) The hexagonal lattices:

FC4 (H f (n, m )) = 0. FC4 (H c (n, m )) = δ1n (m + 1 ). FC4 (H t (n, m )) = δ1n (m + 1 ) + δ1m (n + 1 ). (iii) The triangular lattices:

FC4 (T f (n, m )) = (n − 1 )(m − 1 ) + (n − 2 )(m − 1 ) + (m − 2 )(n − 1 ) = 3nm − 4n − 4m + 5. FC4 (T c (n, m )) = n(m − 1 ) + n(m − 1 ) + n(m − 2 ) + δ4n m = 3nm − 4n + δ4n m. FC4 (T t (n, m )) = 3nm + δ4n m + δ4m n. (iv) The 33 · 42 lattices:

FC4 (S f (n, 2m )) = (n − 1 )(m − 1 ) + (n − 2 )m = 2nm − n − 3m + 1.

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311

FC4 (Sc (n, 2m )) = n(m − 1 ) + nm + 2δ4n m = 2nm − n + 2δ4n m. FC4 (St (n, 2m )) = 2nm + 2δ4n m + δ2m n. 6.3. FC + (G ) 3

First note that, for each 3-cycle uvw in G, the number of C3+ containing uvw is (a − 2 ) + (b − 2 ) + (c − 2 ) = a + b + c − 6 where a, b, c are the degrees of u, v, w. We call uvw a C3 of type (a, b, c) in G. We will proceed by enumerating the number of C3 ’s of each type. (i) Since there is no C3 in square lattices with free boundary condition, we have

FC3+ (Pm Pn ) = 0. For other square lattices we have

FC3+ (Pm Cn ) = 6δ3n (m − 1 ). FC3+ (Cm Cn ) = 6δ3n m + 6δ3m n. (ii) Similarly, since there is no C3 in any hexagonal lattices, we have

FC3+ (H f (n, m )) = FC3+ (H c (n, m )) = FC3+ (H t (n, m )) = 0. (iii) The triangular lattices: • Tf (n, m):

FC3+ (T f (n, m )) = 2 × 4 + 4 × 7 + [2(n − 2 ) + 2(m − 3 )] × 8 + [2(n − 3 ) + 2(m − 3 )] × 10 + 2(n − 3 )(m − 3 ) × 12 = 24nm − 36n − 36m + 52. • Tc (n, m):

FC3+ (T c (n, m )) = 2δ3n × 6 + 2n × 8 + 2n × 10 + (2n(m − 3 ) +δ3n (m − 2 )) × 12 = 24nm − 36n + 12δ3n (m − 1 ). • For Tt (n, m), all triangles are of type (6, 6, 6). Hence

FC3+ (T t (n, m )) = 12(2nm + δ3n m + δ3m n ). (iv) The 33 · 42 lattices: • Sf (n, 2m):

312

H. Lei et al. / Applied Mathematics and Computation 329 (2018) 297–314 Table 1 Summary of enumeration results on the lattice networks. Square lattices Pm Pn

Pm Cn

Cm Cn

WP (G) FP4 (G ) FK1,3 (G ) FC4 (G ) FC3+ (G ) FK4− (G )

6mn − 9m − 9n + 8 18mn − 25m − 25n + 28 4mn − 6m − 6n + 8 (n − 1 )(m − 1 ) 0 0

6mn 18mn 4mn nm + δ4n m + δ4m n 6 δ3 n m + 6 δ3 m n 0

WP (G) FP4 (G ) FK1,3 (G ) FC4 (G ) FC3+ (G ) FK4− (G )

H f (n, m) 9(n + 1 )(m + 1 ) 12(n + 1 )(m + 1 ) 2nm 0 0 0

WP (G) FP4 (G ) FK1,3 (G ) FC4 (G ) FC3+ (G ) FK4− (G )

Tf (n, m) 9nm − 18n − 18m + 31 69nm − 116n − 116m + 185 20nm − 32n − 32m + 50 3nm − 4n − 4m + 5 24nm − 36n − 36m + 52 3nm − 4n − 4m + 5

WP (G) FP4 (G ) FK1,3 (G ) FC4 (G ) FC3+ (G ) FK4− (G )

Sf (n, 2m) 15nm − 13n − 25m + 15 74nm − 46n − 114m + 56 20nm − 12n − 30m + 16 2nm − n − 3m + 1 18nm − 6n − 26m + 6 2nm − 3m

6mn − 9n 18mn − 25n 4mn − 6n n ( m − 1 ) + δ4 n m 6 δ3 n ( m − 1 ) 0 hexagonal lattices H c (n, m) 9m ( n + 1 ) 12m(n + 1 ) 2nm + 2m δ1 n ( m + 1 ) 0 0 triangular lattices Tc (n, m) 9nm − 18n 69nm − 116n 20nm − 32n 3nm − 4n + δ4n m 24nm − 36n + 12δ3n (m − 1 ) 3nm − 4n 33 · 42 lattices Sc (n, 2m) 15nm − 13n 74nm − 46n 20nm − 12n 2nm − n + 2δ4n m (18m − 6 )(n + δ3n ) 2nm

FC3+ (S f (n, 2m )) = 2 × 4 + 2 × 5 + 2(m − 1 ) × 6 + 2(n − 2 ) × 7 + [2(n − 3 ) + 2(m − 2 )] × 8 + 2(n − 3 )(m − 2 ) × 9 = 18nm − 6n − 26m + 6. •

Sc (n,

2m):

FC3+ (Sc (n, 2m )) = 2δ3n × 6 + 2n × 7 + 2n × 8 + (2n(m − 2 ) + δ3n (2m − 2 )) × 9 = (18m − 6 )(n + δ3n ). • For St (n, 2m), all triangles are of type (5, 5, 5). Hence

FC3+ (St (n, 2m )) = 18m(n + δ3n ).

H t (n, m) 9nm − 2 12nm − 2 2(n + 1 )(m + 1 ) δ1 n ( m + 1 ) + δ1 m ( n + 1 ) 0 0 Tt (n, m) 9nm 69nm 20nm 3nm + δ4n m + +δ4m n 12(2nm + δ3n m + δ3m n ) 3nm St (n, 2m) 15nm 74nm 20nm 2nm + 2δ4n m + δ2m n 18m(n + δ3n ) 2nm

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313

6.4. FK − (G ) 4

A K4− contains both C3 and C4 . It is easy to see the following. We omit the proofs. (i) The square lattices:

FK4− (Pm Pn ) = FK4− (Pm Cn ) = FK4− (Cm Cn ) = 0. (ii) The hexagonal lattices:

FK4− (H f (n, m )) = FK4− (H c (n, m )) = FK4− (H t (n, m )) = 0. (iii) The triangular lattices:

FK4− (T f (n, m )) = FC4 (T f (n, m )) = 3nm − 4n − 4m + 5. FK4− (T c (n, m )) = FC4 (T c (n, m )) = 3nm − 4n. FK4− (T t (n, m )) = FC4 (T t (n, m )) = 3nm. (iv) The 33 · 42 lattices:

FK4− (S f (n, 2m )) = m(n − 1 + n − 2 ) = 2nm − 3m. FK4− (Sc (n, 2m )) = 2nm. FK4− (St (n, 2m )) = 2nm. 7. Concluding remarks We studied the numbers of various substructures of order 4 in several common lattice networks. This is an extension of some previous work where the values of some distance-based graph invariants are evaluated for these lattice networks. Our result, together with that in [10], can be summarized in Table 1. These findings provide insights from various aspects on the characteristics of these lattice networks. As mentioned in the introduction, the comparison between the number of length-3 paths and the number of pairs of vertices at distance 3 shows how connected (through length-3 paths) each lattice is. The more “connected” lattice networks, namely the triangular lattices and the 33 · 42 lattices, indeed contain more substructures in general. The fact that these lattice networks generally contain more paths (P4 ) than stars (K1, 3 ) implies that these lattice networks tend to be spread out. Among the lattices under consideration the topology of the triangular lattices seems to be the least spread out. From our proofs it is obvious that the statistics of K1, 3 illustrates the behavior of vertex degrees while that of C3+ appears to reflect the distribution of triangles and their incident edges. Furthermore, between the same lattice networks with different boundary conditions, the difference in the FS (G) function provides useful information on the structural information of each row and column in the lattices. While we attempted to analyze the topologies of these lattices through the numbers of different substructures of the same order, it would be an interesting future project to explore the numbers of substructures of the same type but different sizes. For instance, to study FPk (G ) for k = 5, 6, 7, . . . for each of these lattices. Acknowledgments The authors thank the anonymous referees for their very careful review of an earlier version of this manuscript. Following their valuable suggestions the presentation of the paper was greatly improved. H. Lei and T. Li are partially supported by the Natural Science Foundation of Tianjin (No. 17JCQNJC0 030 0) and the National Natural Science Foundation of China (No. 11771221). T. Li is partially supported by the Natural Science Foundation of Tianjin (No. 16JCYBJC15200). H. Wang is partially supported by the Simons Foundation (#245307). References [1] [2] [3] [4] [5] [6] [7] [8]

A.T. Balaban, Highly discriminating distance based numerical descriptor, Chem. Phys. Lett. 89 (1982) 399–404. A.T. Balaban, Topological indices based on topological distances in molecular graphs, Pure Appl. Chem. 55 (1983) 199–206. ˝ , Graphs of extremal weights, ARS Combin. 50 (1998) 225–233. B. Bollobás, P. Erdos J.A. Bondy, U.S.R. Murty, Graph Theory, Springer, Berlin, 2008. J. Cai, An algorithm for Hamiltonian cycles under implicit degree conditions, ARS Combin. 121 (2015a) 305–313. J. Cai, A sufficient condition involving implicit degree and neighborhood intersection for long cycles, Inf. Process. Lett. 115 (2) (2015b) 225–227. J. Cai, H. Li, A new sufficient condition for pancyclability of graphs, Discret. Appl. Math. 162 (2014) 142–148. S. Cao, M. Dehmer, Y. Shi, Extremality of degree-based graph entropies, Inform. Sci. 278 (2014) 22–33.

314 [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45] [46] [47]

H. Lei et al. / Applied Mathematics and Computation 329 (2018) 297–314 S. Cao, M. Dehmer, Z. Kang, Network entropies based on independent sets and matchings, Appl. Math. Comput. 307 (2017) 265–270. L. Chen, T. Li, J. Liu, Y. Shi, H. Wang, On the wiener polarity index of lattice networks, Plos ONE 11 (2016a) e0167075. L. Chen, J. Liu, Y. Shi, Bounds on the matching energy of unicyclic odd-cycle graphs, MATCH Commun. Math. Comput. Chem. 75 (2016b) 315–330. L. Chen, Y. Shi, The maximal matching energy of tricyclic graphs, MATCH Commun. Math. Comput. Chem. 73 (2015) 105–119. X. Chen, Y. Hou, Some results on Laplacian estrada index of graphs, MATCH Commun. Math. Comput. Chem. 73 (2015) 149–162. Z. Chen, M. Dehmer, Y. Shi, A note on distance-based graph entropies, Entropy 16(10) (2014) 5416–5427. L. da, F. Costa, F. Rodrigues, G. Travieso, P.R.V. Boas, Characterization of complex networks: a survey of measurements, Adv. Phys. 56 (2007) 167–242. H. Deng, The wiener polarity index of molecular graphs of alkanes with a given number of methyl groups, J. Serb. Chem. Soc. 75 (2010) 1405–1412. A.A. Dobrynin, I. Gutman, S. Klavžar, P. Žigert, Wiener index of hexagonal systems, Acta Appl. Math. 72 (2002) 247–294. S. Dorogovtsev, J. Mendes, Evolution of networks, Adv. Phys. 51 (2002) 1079–1187. W. Du, X. Li, Y. Shi, Algorithms and extremal problem on wiener polarity index, MATCH Commun. Math. Comput. Chem. 62 (2009) 235–244. L. Feng, J. Cao, W. Liu, S. Ding, H. Liu, The spectral radius of edge chromatic critical graphs, Linear Algebra Appl. 492 (2016) 78–88. L. Feng, P. Zhang, H. Liu, W. Liu, M. Liu, Y. Hu, Spectral conditions for some graphical propertices, Linear Algebra Appl. 524 (2017a) 182–198. L. Feng, X. Zhu, W. Liu, Wiener index, Harary index and graph properties, Discrete Appl. Math. 223 (2017b) 72–83. M. Ghebleh, A. Kanso, D. Stevanovic, On trees having the same wiener index as their quadratic line graph, MATCH Commun. Math. Comput. Chem. 76 (2016) 731–744. I. Gutman, The energy of a graph, Ber. Math. Stat. Sekt. Forsch. Graz. 103 (1978) 1–22. H. Hua, K.C. Das, On the wiener polarity index of graphs, Appl. Math. Comput. 280 (2016) 162–167. M.H. Khalifeh, H. Yousefi-Azari, A.R. Ashrafi, S.G. Wagner, Some new results on distance-based graph invariants, Eur. J. Comb. 30 (2009) 1149–1163. H. Lei, T. Li, Y. Shi, H. Wang, Wiener polarity index and its generalization in trees, MATCH Commun. Math. Comput. Chem. 78 (1) (2017) 199–212. S. Li, W. Yan, T. Tian, The spectrum and Laplacian spectrum of the dice lattice, J. Stat. Phys. 164 (2016) 449–462. T. Li, H. Dong, Y. Shi, M. Dehmer, A comparative analysis of graph distance measures based on topological indices and graph edit distance, Inform. Sci. 403–404 (2017) 15–21. X. Li, Y. Shi, A survey on the randic´ index, MATCH Commun. Math. Comput. Chem. 59 (2008) 127–156. X. Li, Y. Li, Y. Shi, I. Gutman, Note on the HOMO-LUMO index of graphs, MATCH Commun. Math. Comput. Chem. 70 (1) (2013) 85–96. J. Liu, X. Pan, A unified approach to the asymptotic topological indices of various lattices, Appl. Math. Comput. 270 (2015a) 62–73. J. Liu, X. Pan, Asymptotic incidence energy of lattices, Phys. A 422 (2015b) 193–202. Z. Liang, E. Shan, The clique-transversal set problem in claw-free graphs with degree at most 4, Inf. Process. Lett. 115 (2015) 331–335. J. Liu, X. Pan, Fu. Hu, F. Hu, Asymptotic Laplacian-energy-like invariant of lattices, Appl. Math. Comput. 253 (2015) 205–214. W. Liu, Q. Guo, Y. Zhang, L. Feng, I. Gutman, Further results on the largest matching root of unicyclic graphs, Discret. Appl. Math. 221 (2017) 82–88. J. Ma, Y. Shi, Z. Wang, J. Yue, On wiener polarity index of bicyclic networks, Sci. Rep. 6 (2016) 19066. J. Ma, Y. Shi, J. Yue, The wiener polarity index of graph products, Ars Combin. 116 (2014) 235–244. R. Shrock, F. Wu, Spanning trees on graphs and lattices in d dimensions, J. Phys. A: Math. Gen. 33 (20 0 0) 3881–3902. L.A. Székely, S. Wagner, H. Wang, Problems related to graph indices in trees. recent trends in combinatorics 3c30, IMA Vol. Math. Appl. 159 (2016). Springer, [Cham] S. Wagner, Correlation of graph-theoretical indices, SIAM J. Discret. Math. 21 (1) (2007) 33–46. H. Wiener, Structural determination of paraffin boiling points, J. Am. Chem. Soc. 69 (1947) 17–20. W. Yan, Z. Zhang, Asymptotic energy of lattices, Phys. A 388 (2009) 1463–1471. G. Yu, X. Liu, H. Qu, Singularity of Hermitian (quasi-)Laplacian matrix of mixed graphs, Appl. Math. Comput. 293 (2017) 287–292. G.Y. u, H.Q. u, Hermitian Laplacian matrix and positive of mixed graphs, Appl. Math. Comput. 269 (2015) 70–76. J. Yue, H. Lei, Y. Shi, On the generalized wiener polarity index of trees with a given diameter, Discrete Appl. Math. (2018). In press Y. Zhang, Y. Hu, The Nordhaus-Gaddum-type inequality for the wiener polarity index, Appl. Math. Comput. 273 (2016) 880–884.