Extremal Kirchho index in polycyclic chains Hechao Liu Lihua You School of Mathematical Sciences South China Normal University Guangzhou 510631

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Extremal Kirchhoﬀ index in polycyclic chains

Hechao Liu, Lihua You∗

School of Mathematical Sciences, South China Normal University, Guangzhou, 510631,

P. R. China, e-mail: hechaoliu@m.scnu.edu.cn,ylhua@scnu.edu.cn

∗Corresponding author

Received 12 October 2022

Abstract The Kirchhoﬀ index of graphs, introduced by Klein and Randi´c in 1993,

has been known useful in the study of computer science, complex network and quantum

chemistry. The Kirchhoﬀ index of a graph Gis deﬁned as Kf(G) = P

{u,v}⊆V(G)

ΩG(u, v),

where ΩG(u, v) denotes the resistance distance between uand vin G.

In this paper, we determine the maximum (resp. minimum) k-polycyclic chains with

respect to Kirchhoﬀ index for k≥5, which extends the results of Yang and Klein [Com-

parison theorems on resistance distances and Kirchhoﬀ indices of S, T -isomers, Discrete

Appl. Math. 175 (2014) 87-93], Yang and Sun [Minimal hexagonal chains with respect

to the Kirchhoﬀ index, Discrete Math. 345 (2022) 113099], Sun and Yang [Extremal

pentagonal chains with respect to the Kirchhoﬀ index, Appl. Math. Comput. 437 (2023)

127534] and Ma [Extremal octagonal chains with respect to the Kirchhoﬀ index, arXiv:

2209.10264].

Keywords Kirchhoﬀ index, resistance distance, polycyclic chain.

Mathematics Subject Classiﬁcation: 05C09, 05C12, 05C92

1 Introduction

1.1 Background

Let Gbe a connected graph with vertex set V(G) and edge set E(G). Let dG(u) be

the degree of vertex uin G. The distance between vertex uand vertex vis denoted by

dG(u, v). If we replace each edge of the graph Gwith a unit resistor and regard the graph

Gas an electrical network N, then we deﬁne the eﬀective resistance of vertex uand vertex

vin the electrical network Nas the resistance distance between vertex uand vertex v

in the graph G, and denoted by ΩG(u, v). In this paper, all notations and terminologies

used but not deﬁned can refer to Bondy and Murty [1].

The Wiener index is one of the oldest and most studied topological index from ap-

arXiv:2210.02080v2 [math.CO] 12 Oct 2022

plication and theoretical viewpoints. As a extension of the Wiener index, The Kirchhoﬀ

index is an important measure which contains more information than the Wiener index

and plays an essential role in the research of QSAR and QSPR.

The Wiener index [24] of graph Gis deﬁned as W(G) = P

{u,v}⊆V(G)

dG(u, v), replacing

distance with resistance distance in the deﬁnition of Wiener index, we can obtain the

Kirchhoﬀ index, which is deﬁned as [13]

Kf(G) = X

{u,v}⊆V(G)

ΩG(u, v).

Some mathematical and physical interpretations of Kirchhoﬀ index can be found in [12,14].

The extremal Kirchhoﬀ index had been considered on unicyclic graphs [26], fully loaded

unicyclic graphs [8], cacti [25], graphs with given cut edges [7], graphs with a given vertex

bipartiteness [16], random polyphenyl and spiro chains [9], linear hexagonal (cylinder)

chain [10], generalized phenylenes [15, 32], M¨obius/cylinder octagonal chain [17], linear

phenylenes [19], connected (molecular) graphs [33], and so on.

Some molecular descriptors of polycyclic chains had been considered for many years.

Such as Wiener index [3,5], Kirchhoﬀ index [18,22,27–30], Tutte polynomials [2], Merriﬁed-

Simmons index [4], Kekule structures [23], forcing spectrum [31], k-matching [6], Hosoya

index [21], and so on.

Let Qhbe the linear quadrilateral chain with hsquares and Si(1 ≤i≤h) the i-th

square of Qh. Then the k-polycyclic chain Phcan be obtained from Qhby adding k−4

vertices to Si(1 ≤i≤h) by adding 0 (resp. 1,2,· · · , k −4) vertices to the top edge of Si

(1 ≤i≤h) and the remaining vertices to the bottom edge of Si(1 ≤i≤h). In Figure

1, either D5or L5is a special P5,Z6is a special P6.

For convenience, we suppose that we add dk−4

2evertices to the top edges of S1and

Sh,bk−4

2cvertices to the bottom edges of S1and Sh, and for the Si+1 (1 ≤i≤h−2),

we give a number wi= 0 (resp. 1,2,· · · , k −4) to the k-polygon if the k-polygon is

obtained by adding wivertices to the top edge of Si+1. Then we can use a (h−2)-

vector w= (w1, w2,· · · , wh−2) to denote the k-polycyclic chain, where wi∈ {0,1,· · · , k −

4}. Let Ph(w) (or simply P(w)) be the k-polycyclic chain with h k-polygons and w=

(w1, w2,· · · , wh−2)bea(h−2)-tuple of 0,1,· · · , k −4.

The k-polycyclic chain P(0,0,· · · ,0

| {z }

h−2

) or P(k−4, k −4,· · · , k −4

| {z }

h−2

) is called a helicene

k-polycyclic chain, where P(0,0,· · · ,0

| {z }

h−2

)∼

=P(k−4, k −4,· · · , k −4

| {z }

h−2

), and denoted by

Dh. If k≥6 is even, the k-polycyclic chain P(k−4

2,k−4

2,· · · ,k−4

| {z }

h−2

) is called a lin-

ear k-polycyclic chain, and denoted by Lh. If k≥5 is odd, then k-polycyclic chain

P(bk−4

2c,dk−4

2e,bk−4

2c,dk−4

2e · · ·

| {z }

h−2

) or P(dk−4

2e,bk−4

2c,dk−4

2e,bk−4

2c · · ·

| {z }

h−2

) is

called a zigzag chain, denoted by Zh, where P(bk−4

2c,dk−4

2e,bk−4

2c,dk−4

2e · · ·

| {z }

h−2

)∼

P(dk−4

2e,bk−4

2c,dk−4

2e,bk−4

2c · · ·

| {z }

h−2

). Figure 1 gives D5with k= 6, L5with k= 6

and Z6with k= 7.

Figure 1: D5with k= 6, L5with k= 6 and Z6with k= 7.

1.2 Main results

Our main results are shown as follows.

Theorem 1.1 Let Phbe the set of k-polycyclic chains with h k-polygons (k≥5). Then

for any G∈ Ph, we have

Kf(G)≥Kf(P(0,0,· · · ,0

| {z }

h−2

)),

with equality if and only if G∼

=Dh.

Theorem 1.2 Let Phbe the set of k-polycyclic chains with h k-polygons (k≥5). Then

for any G∈ Ph, we have

Kf(G)≤Kf(P(bk−4

2c,dk−4

2e,bk−4

2c,dk−4

2e,· · ·

| {z }

h−2

)),

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ExtremalKirchhoindexinpolycyclicchainsHechaoLiu,LihuaYouSchoolofMathematicalSciences,SouthChinaNormalUniversity,Guangzhou,510631,P.R.China,e-mail:hechaoliu@m.scnu.edu.cn,ylhua@scnu.edu.cnCorrespondingauthorReceived12October2022AbstractTheKirchhoindexofgraphs,introducedbyKleinandRandicin1993,has...

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Extremal Kirchho index in polycyclic chains Hechao Liu Lihua You School of Mathematical Sciences South China Normal University Guangzhou 510631

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