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Y1 Y2

Figure 2.

2. Resistances between any pairs of vertices in Yn

A ladder graph Ln is a planar graph that looks like a ladder with n rungs. It has 2n vertices and 3n 2 edges. Each of its edges has length 1, so the total length of Ln is − ℓ(Ln) := 3n 2. We obtain the prism graph Yn from Ln by adding two edges connecting the end vertices− on the same side. If we delete an edge that is a rung in Yn, and then contract two edges that are on each side of that rung, we obtain the prism graph Yn 1. If we apply this process to 3-prism − graph Y3, we obtain the graph on the right in Figure 2. We call it 2-prism graph Y2. Similarly, if we apply this process to Y2, we obtain the graph on the left in Figure 2. We call it 1-prism graph Y1. These graphs are the natural extension of prism graphs to the cases n = 1, 2. We see that our formulas for resistance values, Kirchhoff index as well as the spanning tree formulas are also valid for these two cases (see Theorem 3.1 and Equations (10), (16) and (17) below). For a prism graph Yn, we label the vertices on the right and left as q , q , , qn { 1 2 ··· } and p , p , ,pn , respectively. This is illustrated in Figure 3, where 2 i n. We { 1 2 ··· } ≤ ≤ want to find the value of r(p, q) for any two vertices p and q of Yn, where r(x, y) is the resistance function on Yn. We implement the following strategy to do this:

Consider Yn as the union of the ladder graphs Ln i+1 and Li 1 as illustrated in • Figure 4. − − Apply circuit reductions on each of these ladder graphs by keeping the four end • vertices as illustrated in Figure 5. Use our earlier results on a ladder graph in [3] to find the resistances between the • end points of the reduced ladder graphs. Note that certain resistance values are equal to each other because of the symmetries in the ladder graphs. So far we obtain the circuit reduction of Yn by keeping its 8 vertices pn, qn, p , • 1 q1, pi, qi, pi 1 and qi 1. This is illustrated in Figure 6. Find the Moore-Penrose + − − inverse L of the discrete Laplacian matrix L of this reduced Yn. They are 8 8 matrices. × + Use L and Lemma 2.1 to find out the resistances between the 8 vertices of Yn. • Again note that there are symmetries in Yn and that i is an arbitrary value in 2, 3,...,n . { } Symmetries in Yn gives the following identities of resistances:

(1) r(p1,pi)= r(q1, qi) and r(p1, qi)= r(q1,pi), for each i 1,...,n . ∈{ } We recall that the resistance values can be expressed in terms of the entries of the pseudo inverse of the discrete Laplacian matrix. EFFECTIVE RESISTANCES OF PRISM GRAPHS 3

pn p1

q1 qn

Yn

pi-1 pi

qi qi-1

Figure 3.

pn p1

q1 qn

pi-1 pi

qi qi-1 Ln-i+1 Li-1

Figure 4.

Lemma 2.1. [9], [5, Theorem A] Suppose G is a graph with the discrete Laplacian L and the resistance function r(x, y). For the pseudo inverse L+ of L, we have r(p, q)= l+ 2l+ + l+ , for any two vertices p and q of G. pp − pq qq 1 1 1 1 1 1 Let K = 1+ k + m + s and S = 1+ a + b + c , where k, m, s, a, b and c are the resistance values along the edges given in Figure 6. Considering the ordering of the vertices V = p1, pi 1, pi, pn, q1, qi 1, qi, qn , discrete Laplacian matrix L of the graph { − − } (reduced Yn) given in Figure 6 is as follows: K 1 0 1 1 1 0 0 − m − − s − k 1 K 1 0 1 1 0 0  − m − − k − s  0 1 S 1 0 0 1 1  − − a − b − c   1 1 1   1 0 a S 0 0 c b  L =  − − − −  .  1 1 0 0 K 1 0 1   − s − k − m −   1 1 1   0 0 K 1 0   − k − s − m −   1 1 S 1   0 0 b c 0 1 a   − − − −   0 0 1 1 1 0 1 S   − c − b − − a    4 ZUBEYIR CINKIR

pn

qn

pn w qn t u u t p1 w q1 p1

Ln q1 Ln

Figure 5.

p pn 1 b 1 s 1 q1 c qn k c m k a m Yn a

pi-1 p b 1 i s qi 1 qi-1 - + Ln i 1 Li-1

Figure 6.

Then we can compute the Moore-Penrose inverse L+ of L by using [7] with the following formula (see [12, ch 10]):

1 1 1 (2) L+ = L J − + J. − 8 8
where J is of size 8 8 and has all entries 1. Next, we use L+ and Lemma 2.1 to obtain × r(p1,pi 1) and r(p1, qi 1). In this way, we find that − P − 2 − r(p1,pi 1) = Q , where P = m(4bc(ms + k(m + 2s + ms)) + a ((2 + b)(2 + c)ms + k(2(2 + b)(2 + c)s + m(4 + 2c + 4s + cs + b(2 + c + s)))) + 2a(2c(ms + k(m + 2s + ms)) + b(2(1 + c)ms + k(4(1 + c)s + m(2 + 2c + 2s + cs))))) and Q = 2(c(2m + k(2 + m)) + a((2 + c)m + k(2 + c + m)))(b(2s + m(2 + s)) + a((2 + b)s + m(2 + b + s))). R − r(p1,qi 1) = S , where R = k(c(4c(2ms + k(m + s + ms)) + b(4(2 + c)ms + k(2(2 + c)s + m(4 + 2c + 4s + cs)))) + a(b(2 + c)(2(2 + c)ms + k((2 + c)s + m(2 + c + 2s))) + c(4(2 + c)ms + k(2(2 + c)s + m(4 + 2c + 4s + cs))))) and S = 2(c(2m + k(2 + m)) + a((2 + c)m + k(2 + c + m)))(c(2s + k(2 + s)) + b((2 + c)s + k(2 + c + s))). EFFECTIVE RESISTANCES OF PRISM GRAPHS 5 We note that these resistance values can be expressed in a more compact form as below:

1 1 1 r(p1,pi 1)= + , − 1 1 1 1 2  2+ ac + mk ab + ms  a+c + 2+ + m+s (3) m k a b   1 1 1 r(p1, qi 1)= + . − 1 1 1 1 2  2+ ac + mk bc + ks  a+c m+k 2+ b+c k+s   To find the exact values of the resistances in Equation (3), we need the corresponding values from the circuit reduction of Ln i+1 and Li 1. Thus, we turn our attention to the − − circuit reduction of Ln as in Figure 5. Let us apply circuit reductions to a ladder graph Ln by keeping its vertices at its bottom and top so that we obtain a complete graph on 4 vertices. Suppose p1, q1 and pn, qn are the vertices at the bottom and top of the ladder graph. This is illustrated in Figure 5. Note that by the symmetry in Ln, we have only three distinct edge lengths in the complete graph obtained. Let us consider the ordering of the vertices pn, qn, p1, q1 . Using the notations in Figure 5, the Laplacian matrix M of the reduced{ graph can be} given as follows: 1 + 1 + 1 1 1 1 t u w − w − u − t 1 1 + 1 + 1 1 1 M =  − w t u w − t − u  . 1 1 1 + 1 + 1 1  − u − t t u w − w   1 1 1 1 + 1 + 1   − t − u − w t u w   + Then we use symmetries in Ln, compute the Moore-Penrose inverse M of M and apply Lemma 2.1 to write w(wt + u(w +2t)) 1 wt uw r (p , q )= r (p , q )= = + , Ln n n Ln 1 1 2(u + w)(w + t) 2 w + t u + w   u(2wt + u(w + t)) 1 uw ut (4) r (p ,p )= r (q , q )= = + , Ln n 1 Ln n 1 2(u + w)(u + t) 2 u + w u + t   t(2uw +(u + w)t) 1 ut wt r (p , q )= r (q ,p )= = + . Ln n 1 Ln n 1 2(u + t)(w + t) 2 u + t w + t   Using Equation (4), we derive wt An := = rL (pn, qn)+ rL (pn, q ) rL (pn,p ), w + t n n 1 − n 1 uw (5) Bn := = rL (pn, qn) rL (pn, q1)+ rL (pn,p1), u + w n − n n ut Cn := = rL (pn, qn)+ rL (pn, q )+ rL (pn,p ). u + t − n n 1 n 1 As particular cases of Equation (5), we have ac ab bc (6) = Cn i+1, = Bn i+1, = An i+1, a + c − a + b − b + c − mk ms ks (7) = Ci 1, = Bi 1, = Ai 1, m + k − m + s − k + s − where we used the notations as in Equation (3) (and so as in Figure 6). In [3], we gave explicit formulas for the resistance values between any two vertices of a ladder graph. 6 ZUBEYIR CINKIR Next, we use [3, Equation 12 at page 959] to write 2√3 An = 1 √3+ , − − 1 (2 √3)n − − (8) 2√3 Bn = 1 √3+ , − − 1+(2 √3)n − Cn = n 1. − We use Equations (3), (6) and (7) to derive

1 1 1 r(p ,p )= + , 1 i 2 1 1 1 1 2+C − + C 2+B − + B ! (9) n i i n i i 1 1 1 r(p , qi)= + . 1 2 1 + 1 1 + 1 2+Cn−i Ci 2+An−i Ai ! where 2 i n. Next, we use Equation (8) in Equation (10) and then work with [7] to simplify≤ the≤ algebraic expressions as below: n n i+1 i 1 1+ 2 √3 2 √3 − 2 √3 − (n i + 1)(i 1) r(p1,pi)= − − − n− − + − − ,
2√3 1 2
√3
2n (10) − − n  n i+1  i 1 1+ 2 √3 + 2 √ 3 −
+ 2 √3 − (n i + 1)(i 1) r(p1, qi)= − − n − + − − .
2√3 1 2
√3
2n − −   where 1 i n.
≤ ≤ By using the symmetries of the graph Yn, we note that for every i and j in 1, 2,...,n we have { }

(11) r(p1,pi)= r(pj,pk) and r(p1, qi)= r(pj, qk),

where 1 k n and k j + i 1 mod n. Finally,≤ the≤ explicit values≡ of−r(p, q) between any two vertices p and q of the prism graph Yn can be obtained by using Equations (1), (10) and (11). It follows from Equation (10) that n 1 1+ 2 √3 (n i + 1)(i 1) (12) r(p1,pi)+ r(p1, qi)= − n + − − . √3 1 2 √3 n  − −

3. Kirchhoff Index of Yn

In this section, we obtain an explicit formula for Kirchhoff index of Yn by using our explicit formulas derived in 2 for the resistances between any pairs of vertices of Yn. Moreover, we obtain an interesting§ summation formula by combining our findings and what is known in the literature about Kirchhoff index of Yn. Theorem 3.1. For any positive integer n, we have n(n2 1) n2 2 Kf(Yn)= − + 1 . 6 √3 1 (2 √3)n − h − − i EFFECTIVE RESISTANCES OF PRISM GRAPHS 7 Proof. With the notation of vertices as in Figure 3 we have 1 Kf(Y )= r(p, q), by definition [13]. n 2 p,q V (Yn) X∈ n

= n r(p1,pi)+ r(p1, qi), Equations (1) and (11). i=1 X Then the result follows if we use first Equation (12) and do some algebra [7].  Alternatively, we can express the Kirchhoff index formula in Theorem 3.1 as follows: n(n2 1) n2 n Kf(Yn)= − coth ln(2 √3) . 6 − √3 2 −
Kf(Yn) have rational values. For example, its values for 1 n 10 are as follows: 1, 11/3, 47/5, 58/3, 655/19, 279/5, 5985/71, 2540/21, 44193≤ /≤265, 139655/627. Next, we show that an interesting trigonometric sum identity hold: Theorem 3.2. For any positive integer n, we have n 1 − 1 n 2 = 1 . 2 kπ √ √ n − k 1 + 2 sin ( n ) 3 1 (2 3) X=0 h − − i Proof. Prism graph Yn can be seen as the cartesian product P2Cn, where P2 is the path graph with 2 vertices and Cn is the cycle graph with n vertices. Moreover, considering the Laplacian eigenvalues of P2 and Cn we see that the Laplacian eigenvalues of Yn (see [8], [10] and [11]) are iπ 2jπ (13) λij =4 2cos( ) 2cos( ), where i =0, 1 and j =0, 1,...,n 1. − 2 − n − We recall that [14, pg 644] n 1 − 1 n2 1 (14) = − . 2 kπ 3 k sin ( n ) X=1 Now, we can express the Kirchhoff index via the eigenvalues of the discrete Laplacian matrix of Yn [13]: (15) 1 Kf(Yn)=2n λij λij =0 X6 n 1 n 1 − 1 − 1 = n 2kπ + n 2kπ , by Equation (13), k 1 cos( n ) k 2 cos( n ) X=1 − X=0 − n 1 n 1 − 1 − 1 2kπ kπ = n + n , using 1 cos( ) = 2 sin2 ( ), 2 kπ 2 kπ − n n k 2 sin ( n ) k 1 + 2 sin ( n ) X=1 X=0 n 1 n(n2 1) − 1 = − + n , by Equation (14). 6 2 kπ k 1 + 2 sin ( n ) X=0 Then the proof is completed by combining Equation (15) and the result in Theorem 3.1.  8 ZUBEYIR CINKIR 4. Recursive Formulations In this section, we give recursive formulas for the resistance values obtained in 2, the § Kirchhoff index of Yn and the trigonometric formula given in Theorem 3.2. As we did in [3] for Ladder graph, we use the sequence of integers Gn defined by the following recurrence relation Gn =4Gn Gn, if n 2, and G = 0, G =1. +2 +1 − ≥ 0 1 We have n n (2 √3)− (2 √3) Gn = − − − , for each integer n 0. 2√3 ≥ The sequence Gn has various well-known properties [15]. For example, it gives the number of spanning trees of Ln [2], and the number of the spanning trees of the prism graph Yn [6] is given by n n G (16) (2 + √3)n + (2 √3)n 2 = 2n 2 . 2 − − 2 Gn − n  1    Let gn = (2 √3) = for any nonnegative integer n. Then, for any integer − Gn+1 (2 √3)Gn i 1, 2,...,n we have − − ∈{ } 2 2 (n i + 1)(i 1) Gn 1 Gn r(p1,pi)= − − + + (gn i+1 + gi 1), 2n G2n 2Gn − 4√3 2G2n 4Gn − − −2  −2  (n i + 1)(i 1) Gn 1 Gn r(p1, qi)= − − + + + (gn i+1 + gi 1). 2n G2n 2Gn 4√3 2G2n 4Gn − − −  −  Similarly, the other resistance values can be expressed in terms of Gn by using the sym- metries in Yn like Equation (1) and Equation (11). Here are how we can express the results given in Theorem 3.1 and Theorem 3.2 in terms of Gn: 2 2 2 n 1 2 n(n 1) 2n Gn − 1 2nGn (17) Kf(Yn)= − + and 2 kπ = . 6 G n 2Gn G n 2Gn 2 k 1 + 2 sin ( n ) 2 − X=0 − Acknowledgements: This work is supported by Abdullah Gul University Foundation of Turkey. References [1] R. B. Bapat and S. Gupta, Resistance distance in wheels and fans Indian J. Pure Appl. Math., 41(1), (2010), 1–13. [2] F. T. Boesch and H. Prodinger, Spanning tree formulas and Chebyshev polynomials, Graphs and Combinatorics, Volume 2, Issue 1, (1986), 191–200. [3] Z. C¸ınkır, Effective Resistances and Kirchhoff index of Ladder Graphs, Journal of Mathematical Chemistry, Volume 54, Number 4, (2016), 955–966. [4] A. Carmona, A. M. Encinas and M. Mitjana, Effective resistances for ladder-like chains, Int. J. Quantum Chem., Vol 114, (2014), 1670-1677. [5] D. J. Klein and M. Randi´c, Resistance distance, Journal Mathematical Chemistry, 12 (1993) 81–95. [6] Jagers, A. A., A note on the number of spanning trees in a prism graph, International Journal of Computer Mathematics, 24 (2): (1988), 151-154. [7] Wolfram Research, Inc., Mathematica, Version 9.0, Wolfram Research Inc., Champaign, IL., (2012). [8] J.B. Liu, J. Cao, A. Alofi, A. Al-Mazrooei, A. Elaiw, Applications of Laplacian Spectra for n-Prism Networks, Neurocomputing, Volume 198, (2016), 69–73. [9] R.B. Bapat, Resistance matrix of a weighted graph, MATCH Commun. Math. Comput. Chem. 50 (2004), 73–82. [10] R.B. Bapat, Graphs and Matrices, Springer-Verlag, London, 2014. EFFECTIVE RESISTANCES OF PRISM GRAPHS 9 [11] D. M. Cvetkovi´c, M. Doob, H. Sachs, Spectra of Graphs, Theory and Application, Academic Press, New York, 1980. [12] C. Rao and S. Mitra, Generalized Inverse of Matrices and Its Applications, John Wiley and Sons, 1971. [13] D. J. Klein and M. Randi´c, Resistance distance, Journal Mathematical Chemistry, 12, (1993), 81-95. [14] A. P. Prudnikov, Y. A. Brychkov and O. I. Marickev, Integral and Series, Volume 1: Elementary Functions, Gordon & Breach, New York, (1998). [15] N. J. A. Sloane. The On-Line Encyclopedia of Integer Sequences, http://oeis.org. Sequence A001353.

Zubeyir Cinkir, Department of Industrial Engineering, Abdullah Gul¨ University, Kay- seri, TURKEY. E-mail address: [email protected]