A note on polyomino chains with extremum general sum-connectivity

A note on polyomino chains with extremum general sum-connectivity index

arXiv:1803.04657v1 [math.CO] 13 Mar 2018

Akbar Ali∗, Tahir Idrees Knowledge Unit of Science, University of Management & Technology, Sialkot-Pakistan E-mail: [email protected], [email protected]

Submitted on January 2, 2018

Abstract P The general sum-connectivity index of a graph G is defined as χα (G) = uv∈E(G) (du + dv )α where du is degree of the vertex u ∈ V (G), α is a real number different from 0 and uv is the edge connecting the vertices u, v. In this note, the problem of characterizing the graphs having extremum χα values from a certain collection of polyomino chain graphs is solved for α < 0. The obtained results together with already known results (concerning extremum values of polyomino chain graphs) give the complete solution of the aforementioned problem.

1

Introduction

All graphs considered in this note are simple, finite and connected. Those notations and terminologies from graph theory which are not defined here can be found in the books [12, 23]. The connectivity index (also known as Randić index and branching index) is one of the most studied graph invariants, which was introduced in 1975 within the study of molecular branching [31]. The connectivity index for a graph G is defined as X 1 R(G) = (du dv )− 2 , uv∈E(G)

where du represents the degree of the vertex u ∈ V (G) and uv is the edge connecting the vertices u, v of G. Detail about the mathematical properties of this index can be found in the survey [26], recent papers [1, 8, 16, 17, 20, 27, 29] and related references contained therein. Several modified versions of the connectivity index were appeared in literature. One of such versions is the sum-connectivity index [36], which is defined as X 1 χ(G) = (du + dv )− 2 . uv∈E(G)

Soon after the appearance of sum-connectivity index, its generalized version was proposed [37], whose definition is given as X χα (G) = (du + dv )α , uv∈E(G)

where α is a non-zero real number. In this note, we are concerned with the general sumconnectivity index χα . Details about χα can be found in the recent papers [3, 4, 7, 11, 15, 25, 30, ∗

Corresponding author.

1

32, 33, 38] and related references listed therein. We recall that 2χ−1 (G) = H(G), where H is the harmonic index [19], and χ1 coincides with the first Zagreb index [21], whose mathematical properties can be found in the recent surveys [9, 13, 14] and related references cited therein. A polyomino system is a connected geometric figure obtained by concatenating congruent squares side to side in a plane in such a way that the figure divides the plane into one infinite (external) region and a number of finite (internal) regions, and all internal regions must be congruent squares. For possible applications of polyomino systems, see, for example, [22, 24, 28, 35] and related references mentioned therein. Two squares in a polyomino system are adjacent if they share a side. A polyomino chain is a polyomino system in which every square is adjacent to at most two other squares. Every polyomino chain can be represented by a graph known as polyomino chain graph. For the sake of simplicity, in the rest of this note, by the term polyomino chain we always mean polyomino chain graph. The problem of characterizing graphs having extremum χα values over the collection of certain polyomino chains, with fixed number of squares, was solved in [5, 6, 34] for α = 1. The results established in [18] give a solution of the aforementioned problem for α = −1. An and Xiong [2] solved this problem for α > 1. While, the same problem was also addressed in [10] and its solution for the case 0 < α < 1 was reported there. The main purpose of the present note is to give the solution of the problem under consideration for all remaining values of α, that is, for α < −1 and −1 < α < 0.

2

Main Results

Before proving the main results, we recall some definitions concerning polyomino chains. In a polyomino chain, a square adjacent with only one (respectively two) other square(s) is called terminal (respectively non-terminal) square. A kink is a non-terminal square having a vertex of degree 2. A polyomino chain without kinks is called linear chain. A polyomino chain consisting of only kinks and terminal squares is known as zigzag chain. A segment is a maximal linear chain in a polyomino chain, including the kinks and/or terminal squares at its ends. The number of squares in a segment Sr is called its length and is denoted by l(Sr ) (or simply by lr ). If a polyomino chain Bn has segments S1 , S2 , ..., Ss then the vector (l1 , l2 , ..., ls ) is called length vector of Bn . A segment Sr is said to be external (internal, respectively) segment if Sr contains (does not contain, respectively) terminal square. Definition 2.1. [34] For 2 ≤ i ≤ s − 1 and 1 ≤ j ≤ s, ( 1 if li = 2 αi = 0 if li ≥ 3 βj =

(

1 if lj = 2 0 if lj ≥ 3

and α1 = αs = 0. Let Ωn be the collection of all those polyomino chains, having n squares, in which no internal segment of length 3 has edge connecting the vertices of degree 3. Theorem 2.1. [10] Let Bn ∈ Ωn be a polyomino chain having s segment(s) S1 , S2 , S3 , ..., Ss with the length vector (l1 , l2 , ..., ls ). Then, χα (Bn ) = 3 · 6α n + (2 · 5α − 6α+1 + 4 · 7α )s + (2 · 4α + 2 · 5α + 6α − 4 · 7α ) s X αi . + (2 · 6α − 5α − 7α )[β1 + βs ] + (5 · 6α − 2 · 5α − 4 · 7α + 8α ) i=1

2

Let f (α) = 2 · 5α − 6α+1 + 4 · 7α , g(α) = 2 · 6α − 5α − 7α , h(α) = 5 · 6α − 2 · 5α − 4 · 7α + 8α . Furthermore, let Ψχα (S1 ) = f (α) + g(α)β1 , Ψχα (Ss ) = f (α) + g(α)βs and for s ≥ 3, assume that Ψχα (Si ) = f (α) + h(α)αi where 2 ≤ i ≤ s − 1. Then Ψχα (Bn ) =

s X

Ψχα (Si ) = f (α)s + g(α)(β1 + βs ) + h(α)

s X

αi .

(1)

i=1

i=1

Hence, the formula given in Theorem 2.1 can be rewritten as χα (Bn ) = 3 · 6α n + (2 · 4α + 2 · 5α + 6α − 4 · 7α ) + Ψχα (Bn ).

(2)

The next lemma is a direct consequence of the relation (2). Lemma 2.2. [10] For any polyomino chain Bn having n ≥ 3 squares, χα (Bn ) is maximum (respectively minimum) if and only if Ψχα (Bn ) is maximum (respectively minimum). Lemma 2.2 will play a vital role in proving the main results of the present note. Lemma 2.3. [10] Let Bn ∈ Ωn be a polyomino with n ≥ 3 squares. If f (α), f (α) + 2g(α) and f (α) + 2h(α) are all negative, then χα (Zn ) ≤ χα (Bn ) ≤ χα (Ln ). Right (respectively left) equality holds if and only if Bn ∼ = Ln (respectively Bn ∼ = Zn ). Proposition 2.4. Let Bn ∈ Ωn be a polyomino chain having n ≥ 3 squares. Let x0 ≈ −3.09997 be a root of the equation f (α) = 0. Then, for x0 < α < 0, it holds that χα (Zn ) ≤ χα (Bn ) ≤ χα (Ln ), with right (respectively left) equality if and only if Bn ∼ = Ln (respectively Bn ∼ = Zn ). Proof. It can be easily checked that f (α), f (α) + 2g(α) and f (α) + 2h(α) are negative for x0 < α < 0, and hence, from Lemma 2.3, the required result follows. Proposition 2.5. Let Bn ∈ Ωn be a polyomino with n ≥ 3 squares. Let x0 ≈ −3.09997 be a root of the equation f (α) = 0. Then, for α ≤ x0 , the following inequality holds χα (Bn ) ≥ χα (Zn ), with equality if and only if Bn ∼ = Zn . Proof. We note that f (α) is non-negative and both g(α), h(α) are negative for α ≤ x0 ≈ −3.09997. Suppose that the polyomino chain Bn∗ ∈ Ωn has the minimum Ψχα value for α ≤ x0 . Further suppose that S1 , S2 , ..., Ss be the segments of Bn∗ with the length vector (l1 , l2 , ..., ls ). It holds that Ψχα (Zn ) = 2f (α) + 2g(α) + (n − 3)(f (α) + h(α)) ≤ 2f (α) + 2g(α) < f (α) = Ψχα (Ln ), which implies that s ≥ 2. If at least one of external segments of Bn∗ has length greater than 2. Without loss of generality, (1) assume that l1 ≥ 3. Then, there exist a polyomino chain Bn ∈ Ωn having length vector ( 2, 2, · · · , 2 , l2 , ..., ls ) and | {z } (l1 −1)−times

Ψχα (Bn(1) ) − Ψχα (Bn∗ ) = g(α) + (l1 − 2) (f (α) + h(α)) ≤ f (α) + g(α) + h(α) < 0, 3

for α ≤ x0 ≈ −3.09997, which is a contradiction to the definition of Bn∗ . Hence both external segments of Bn∗ must have length 2. If some internal segment of Bn∗ has length greater than 2, say lj ≥ 3 where 2 ≤ j ≤ s − 1 and (2) s ≥ 3. Then, there exists a polyomino chain Bn ∈ Ωn having length vector (l1 , l2 , ..., lj−1 , 2, lj − 1, ..., ls ) and Ψχα (Bn(2) ) − Ψχα (Bn∗ ) = f (α) + (1 + y)h(α) < 0,

(where y = 0 or 1)

for α ≤ x0 ≈ −3.09997, which is again a contradiction. Hence, every internal segment of Bn∗ has length 2. Therefore, Bn∗ ∼ = Zn and from Lemma 2.2, the desired result follows. Proposition 2.6. Let Bn ∈ Ωn be a polyomino with n ≥ 3 squares. Let α ≈ −3.09997 be a root of the equation f (α) = 0. Then, the following inequality holds χα (Bn ) ≤ 3 · 6α n + (2 · 4α + 2 · 5α + 6α − 4 · 7α ), with equality if and only if Bn does not contain any segment of length 2. Proof. From Equation (1), it follows that Ψχα (Bn ) = g(α)(β1 + βs ) + h(α)

s X

αi ≤ 0.

i=1

Clearly, the equality Ψχα (Bn ) = 0 holds if and only if Bn does not contain any segment of length 2. Hence, by using Lemma 2.2, we have the required result. Let Zn∗ be a subclass of Ωn consisting of those polyomino chains which do not contain any segment of length equal to 2 or greater than 4, and contain at most one segment of length 4. Let Zn be a subclass of Ωn consisting of those polyomino chains in which every internal segment (if exists) has length 3 or 4, every external segment has length at most 4, at most one external segment has length 2, at most one segment has length 4 and if some internal segment has length 4 then both the external segments have length 3. Let Zn† ∈ Ωn be the polyomino chain in which every internal segment (if exists) has length 3, every external segment has length at most 3 and at most one external segment has length 2. Proposition 2.7. Let Bn ∈ Ωn be a polyomino with n ≥ 3 squares. Let x0 ≈ −3.09997 and x1 ≈ −5.46343 be the roots of the equations f (α) = 0 and f (α) + g(α) = 0, respectively. Then, for x1 < α < x0 , the following inequality holds χα (Bn ) ≤ χα (Zn∗ ) ,

(3)

with equality if and only if Bn ∼ = Zn∗ ∈ Zn∗ . Also, for α = x1 , the following inequality holds χα (Bn ) ≤ χα Znz ,

(4)

with equality if and only if Bn ∼ = Znz ∈ Zn . Furthermore, for α < x1 , the following inequality holds (5) χα (Bn ) ≤ χα Zn† ,

† with equality if and only if Bn ∼ = Zn .

4

Proof. For n = 3, the result is obvious. We assume that n ≥ 4. It can be easily checked that f (α) is positive and both g(α), h(α) are negative for x1 < α < x0 . Suppose that for the polyomino chain Bn∗ ∈ Ωn , Ψχα (Bn∗ ) is maximum for α < x0 . Let Bn∗ has s segments S1 , S2 , ..., Ss with the length vector (l1 , l2 , ..., ls ). If s ≥ 3 and at least one of internal segments of Bn∗ has length 2, say li = 2 for 2 ≤ i ≤ s − 1, (1) then there exists a polyomino chain Bn ∈ Ωn having length vector ( (l1 , l2 , ..., ls−1 + ls − 1) (l1 , l2 , ..., li−1 , li + li+1 − 1, li+2 , ..., ls ) and Ψχα (Bn∗ ) −

Ψχα (Bn(1) )

if i = s − 1, otherwise,

( f (α) + x · g(α) + h(α) < 0 = f (α) + (1 + y)h(α) < 0

if i = s − 1, otherwise,

for α < x0 , where x, y ∈ {0, 1}. This is a contradiction. Hence, every internal segment (if exists) of Bn∗ has length greater than 2. If at least one of segments of Bn∗ has length greater than 4, say li ≥ 5 for 1 ≤ i ≤ s, then (2) there exists a polyomino chain Bn ∈ Ωn having length vector

and

  (3, l1 − 2, l2 , l3 , ..., ls ) (l1 , l2 , ..., li−1 , 3, li − 2, li+1 , li+2 , ..., ls )   (l1 , l2 , ..., ls−1 , 3, ls − 2)

if i = 1, if 2 ≤ i ≤ s − 1, if i = s,

Ψχα (Bn∗ ) − Ψχα (Bn(2) ) = −f (α) < 0, a contradiction. Hence, every segment of Bn∗ has length less than than 5. If at least two segments of Bn∗ have length 4, say li = lj = 4 for 1 ≤ i, j ≤ s, then there (3) exists a polyomino chain Bn ∈ Ωn having length vector (3, l1 , l2 , ..., li−1 , li − 1, li+1 , ..., lj−1 , lj − 1, lj+1 , ..., ls ) and Ψχα (Bn∗ ) − Ψχα (Bn(3) ) = −f (α) < 0, a contradiction. Hence, Bn∗ contains at most one segment of length 4. If both the external segments of Bn∗ have length 2, then (s ≥ 3 because n ≥ 4) there exists (4) a polyomino chain Bn ∈ Ωn having length vector (l1 + 1, l2 , l3 , ..., ls−1 ) and Ψχα (Bn∗ ) − Ψχα (Bn(4) ) = f (α) + 2g(α) < 0, which is again a contradiction. Hence, at most one external segment has length 2. In what follows, without loss of generality, we assume that ls = 2 whenever some external segment has length 2. If some external segment of Bn∗ has length greater 2, say l1 = 2, then there exists a polyomino (5) chain Bn ∈ Ωn having length vector (l2 + 1, l3 , l4 , ..., ls ) and Ψχα (Bn∗ ) − Ψχα (Bn(5) ) = f (α) + g(α) < 0, (because l2 ≥ 3) for x1 < α < x0 , which is again a contradiction. Hence, if Ψχα (Bn∗ ) is maximum for x1 < α < x0 then every external segment of Bn∗ has length greater than 2. Therefore, if Ψχα (Bn∗ ) is maximum for x1 < α < x0 then Bn∗ ∼ = Zn∗ and thence from Lemma 2.2, inequality (3) follows. In the remaining proof, we assume α ≤ x1 . 5

If Bn∗ contains a segment of length 4, say li = 4 for (6) chain Bn ∈ Ωn having length vector   (2, l1 − 1, l2 , l3 , ..., ls )    (l , l , ..., l , l − 1, l , l , ..., l + 1) 1 2 i−1 i i+1 i+2 s (l1 , l2 , ..., li−1 , li − 1, li+1 , li+2 , ..., ls , 2)    (l , l , ..., l , l − 1, 2) 1 2 s−1 s

and

( g(α) Ψχα (Bn∗ ) − Ψχα (Bn(6) ) = −f (α) − g(α)

1 ≤ i ≤ s, then there exists a polyomino

if if if if

i = 1, 2 ≤ i ≤ s − 1 and ls = 2, 2 ≤ i ≤ s − 1 and ls = 3, i = s,

if 2 ≤ i ≤ s − 1 and ls = 2, otherwise.

This last equation together with the fact that for α < x1 , both g(α) and −f (α) − g(α) are negative, gives a contradiction. The same equation together with the fact that for α = x1 , only g(α) is negative, arises also a contradiction if 2 ≤ i ≤ s − 1 and ls = 2. Therefore, if Ψχα (Bn∗ ) is † maximum for α < x1 then Bn∗ ∼ = Zn and if Ψχα (Bn∗ ) is maximum for α = x1 then Bn∗ ∈ Zn , and thence from Lemma 2.2, inequalities (4) and (5) follow. Propositions 2.4, 2.5, 2.6 and 2.7, together with the already reported results in [2, 5, 6, 10, 18, 34], yield Table 1 which gives information about the polyomino chains having extremum χα values in the collection Ωn for n ≥ 3. α>0 x0 < α < 0 α = x0 x1 < α < x0 α = x1 α < x1

Polyomino Chain(s) with Maximal χα Value Zn Ln chains having no segment of length 2 members of Zn∗ members of Zn Zn†

Polyomino Chain(s) with Minimal χα Value Ln Zn Zn Zn Zn Zn

Table 1: Polyomino chains having extremum χα values in the collection Ωn for n ≥ 3.

References [1] M. An, L. Xiong, Extremal polyomino chains with respect to general Randić index, J. Comb. Optim. 31 (2016) 635-647. [2] M. An, L. Xiong, Extremal polyomino chains with respect to general sum-connectivity index, Ars Combinatoria 131 (2017) 255–271. [3] S. Akhter, M. Imran, Z. Raza, Bounds for the general sum-connectivity index of composite graphs, J. Inequal. Appl. 2017 (2017) 76. [4] A. Ali, An alternative but short proof of a result of Zhu and Lu concerning general sumconnectivity index, Asian-European J. Math. 11 (2018) DOI: 10.1142/S1793557118500304, in press. [5] A. Ali, A. A. Bhatti, Z. Raza, A note on the zeroth-order general Randić index of cacti and polyomino chains, Iranian J. Math. Chem. 5 (2014) 143-152. [6] A. Ali, A. A. Bhatti, Z. Raza, Some vertex-degree-based topological indices of polyomino chains, J. Comput. Theor. Nanosci. 12 (2015) 2101-2107. [7] A. Ali, D. Dimitrov, On the extremal graphs with respect to bond incident degree indices, Discrete Appl. Math. 238 (2018) 32–40. [8] A. Ali, Z. Du, On the difference between atom-bond connectivity index and Randić index of binary and chemical trees, Int. J. Quantum Chem. 117 (2017) e25446. 6

[9] A. Ali, I. Gutman, E. Milovanović, I. Milovanović, Sum of powers of the degrees of graphs: extremal results and bounds, MATCH Commun. Math. Comput. Chem. 80 (2018) 5–84. [10] A. Ali, Z. Raza, A. A. Bhatti, Bond incident degree (BID) indices of polyomino chains: A unified approach, Appl. Math. Comput. 287-288 (2016) 28–37. [11] M. Arshad, I. Tomescu, Maximum general sum-connectivity index with −1 ≤ α < 0 for bicyclic graphs, Math. Reports 19 (2017) 93–96. [12] J. A. Bondy, U. S. R. Murty, Graph Theory, Springer, London, 2008. [13] B. Borovićanin, K. C. Das, B. Furtula, I. Gutman, Bounds for Zagreb indices, MATCH Commun. Math. Comput. Chem. 78 (2017) 17–100. [14] B. Borovićanin, K. C. Das, B. Furtula, I. Gutman, Zagreb indices: Bounds and extremal graphs, in: I. Gutman, B. Furtula, K. C. Das, E. Milovanović, I. Milovanović (Eds.), Bounds in Chemical Graph Theory – Basics, Univ. Kragujevac, Kragujevac, 2017, pp. 67–153. [15] Q. Cui, L. Zhong, On the general sum-connectivity index of trees with given number of pendent vertices, Discrete Appl. Math. 222 (2017) 213–221. [16] Q. Cui, L. Zhong, The general Randić index of trees with given number of pendent vertices, Appl. Math. Comput. 302 (2017) 111–121. [17] K. C. Das, S. Balachandran, I. Gutman, Inverse degree, Randić index and harmonic index of graphs, Appl. Anal. Discr. Math. 11 (2017) 304–313. [18] H. Deng, S. Balachandran, S. K. Ayyaswamy, Y. B. Venkatakrishnan, The harmonic indices of polyomino chains, Natl. Acad. Sci. Lett. 37 (2014) 451-455. [19] S. Fajtlowicz, On conjectures of Graffiti-II, Congr. Numer. 60 (1987) 187–197. [20] I. Gutman, B. Furtula, V. Katanić, Randić index and information, AKCE Int. J. Graphs Comb. DOI:10.1016/j.akcej.2017.09.006, in press. [21] I. Gutman, N. Trinajstić, Graph theory and molecular orbitals. Total π-electron energy of alternant hydrocarbons, Chem. Phys. Lett. 17 (1972) 535–538. [22] I. Gutman, S. Klavzar, A. Rajapakse, Average distances in square–cell configurations. Int. J. Quant. Chem. 76 (1999) 611–617. [23] F. Harary, Graph Theory, Addison–Wesley, Reading, 1969. [24] F. Harary, P.G. Mezey, Cell–shedding transformations, equivalence relations, and similarity measures for square–cell configurations. Int. J. Quant. Chem. 62 (1997) 353–361. [25] M. K. Jamil, I. Tomescu, Minimum general sum-connectivity index of trees and unicyclic graphs having a given matching number, Discrete Appl. Math. 222 (2017) 143–150. [26] X. Li, Y. Shi, A survey on the Randić index, MATCH Commun. Math. Comput. Chem. 59 (2008) 127–156. [27] F. Li, Q. Ye, The general connectivity indices of fluoranthene-type benzenoid systems, Appl. Math. Comput. 273 (2016) 897–911. [28] S. Liu, J. Ou, On maximal resonance of polyomino graphs, J. Math. Chem. 51 (2013) 603–619. [29] T. Mansour, M. A. Rostami, S. Elumalai, B. A. Xavier, Correcting a paper on the Randić and geometric-arithmetic indices, Turk. J. Math. 41 (2017) 27–32. [30] H. S. Ramane, V. V. Manjalapur, I. Gutman, General sum-connectivity index, general product-connectivity index, general Zagreb index and coindices of line graph of subdivision graphs, AKCE Int. J. Graphs Comb. 14 (2017) 92–100. 7

[31] M. Randić, On characterization of molecular branching, J. Am. Chem. Soc. 97 (1975) 6609–6615. [32] I. Tomescu, On the general sum-connectivity index of connected graphs with given order and girth, Electron. J. Graph Theory Appl. 4 (2016) 1–7. [33] H. Wang, J.-B. Liu, S. Wang, W. Gao, S. Akhter, M. Imran, M. R. Farahani, Sharp bounds for the general sum-connectivity indices of transformation graphs, Discrete Dyn. Nat. Soc. 2017 (2017) 2941615. [34] Z. Yarahmadi, A. R. Ashrafi, S. Moradi, Extremal polyomino chains with respect to Zagreb indices, Appl. Math. Lett. 25 (2012) 166-171. [35] L. Zhang, S. Wei, F. Lu, The number of Kekulé structures of polyominos on the torus, J. Math. Chem. 51 (2013) 354–368. [36] B. Zhou, N. Trinajstić, On a novel connectivity index, J. Math. Chem. 46 (2009) 1252– 1270. [37] B. Zhou, N. Trinajstić, On general sum-connectivity index, J. Math. Chem. 47 (2010) 210–218. [38] Z. Zhu, H. Lu, On the general sum-connectivity index of tricyclic graphs, J. Appl. Math. Comput. 51 (2016) 177–188.

8