# Computing Connection-Based Topological Indices of Sudoku Graphs

Authors: Muhammad Basit Ali; Ebenezer Bonyah; Muhammad Javaid
Journal: Journal of Mathematics
 (2022)

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## Abstract

A topological index (TI) is a function that associates a numeric number to the under-study molecular graph. The first TI based on distance was presented by Wiener to find the boiling point of paraffins. In 1972, to compute the totalπ-electron energy of a molecule, the degree-based indices are used. In this paper, we obtained first and second Zagreb connection, modified first, second, third, and fourth Zagreb connection, first, second, third, and fourth multiplicative Zagreb connection, and modified first, second, and third multiplicative Zagreb connection indices for Sudoku graphs.


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## Body

## 1. Introduction

The field of graph theory is comprehensively growing and assuming an exceptional part in the subject of cheminformatics that is the combination of chemistry, mathematics, and information technology which studies the diverse chemical formation and their physicochemical properties. In particular, Graph theory is used in a branch of mathematical chemistry which is known as chemical graph theory [17, 29]. Chemical graph theory gives a stage to concentrate on the physicochemical properties of the molecular graph with the assistance of topological indices (TIs). TIs are the numerical numbers which are associated with different chemical structures of molecular graphs and anticipate the structural, toxicological, biological, and physicochemical properties of the chemical compounds existing in the molecular graphs [31]. Moreover, TIs are same for the isomorphic graphs. Up till now, many researchers have worked on TIs, see [21, 32]. A remarkable utilization of these TIs has been generally displayed in the investigation of qualitative structural activity and property relationships (QSAR and QSPR). Todeschini and Consonni showed that the connection numbers are richly utilized in the subject of cheminformatics, see [13, 18, 26].There are different types of TIs such as distance-based TIs, degree-based TIs, and polynomial-related TIs. Among these, the degree-based topological indices are of remarkable importance and expect to be a key part in chemical graph theory and particularly in chemistry [16]. This class is further grouped into two subclasse degrees and connection-based TIs. These TIs were characterized by Gutman and Trinajsti c´ [14] to derive the whole π-electron energy of alternant hydrocarbons. For more review, we refer to [7, 9, 25]. Gutman, Rucic, and Furtula also worked on various TIs, see [12, 15]. In 2003, Nikoli c´ [24] described modified Zagreb indices. In 2011, Hao [19] utilized modified approach and accomplished the trial of Zagreb and modified Zagreb indices. Dhanalakshmi [7] derived many results for modified and multiplicative ZIs of graph obtained by different operators. In 2013, Das [8] derived upper bounds for the multiplicative Zagreb indices of operations on graphs. Many researchers work on multiplicative Zagreb indices [10, 11] and sum-multiplicative Zagreb indices [27, 33].Connection number is a number of such vertices which are on distance 2 from a specific vertex. Trinajstic [1] studied the modification of first Zagreb indices. Tang [28] fostered certain Zagreb connection indices (ZCIs). Moreover, Ali [2] derived modified ZCIs of T-sum graphs. In recent times, Liu [23] computed ZCIs of molecular graphs based on operations. Cao [4] obtained ZCIs to work out both exact and upper bounds of some product-related graphs. Moreover, Javaid et al. [22] found out novel connection-based ZIs of numerous wheel related graphs, Javaid et al. [30] also derived connection-based multiplicative Zagreb Indices of dendrimer nanostars, and Haoer [20] derived the multiplicative leap ZIs.In this paper, we constructed the partitions of the edges of the Sudoku graphs with respect to their connection numbers (see Table1) and degree-connection numbers (see Table 2) of their end vertices. Moreover, the partition of the vertex set of the Sudoku graph is obtained with respect to the degree and connection number considering at the same time (see Table 3). Finally, we obtained first and second Zagreb connection, modified first, second, third, and fourth Zagreb connection, first, second, third, and fourth multiplicative Zagreb connection, and modified first, second, and third multiplicative Zagreb connection indices that are computed for Sudoku graphs. This paper is composed as follows: Section 2 presents a few significant definitions which are required to lay out the concept of the present study. Section 3 is about the construction of Sudoku graph, Section 4 and Section 5 consist of the main results, and Section 6 covers the conclusion.Table 1
Connection number-based partition of edges ofSKn×n.
Eλn1,λn2cEλn1,λn2cEλn1,λn2cEλn1,λn2cE5,6c12E6,9c24E9,11c16E11,14c8E11,6c16E6,14c12E6,6c12E11,5c8E11,11c4E10,13c28n−16E13,15c28n−16E14,15c8n−1E9,10c82n−3E9,13c28n−16E9,15c28n−16E9,14c4n−8E9,9c4n−8E10,14c8p−16E13,14c8n−16E16,16c26n2−110n+116E12,16c8n−22E11,13c8E15,16c8n−16E14,16c4n−8E10,10c4n−3E13,13c4n−3E15,15c4n−3Table 2
Degree-connection number-based partition of edges ofSKn×n.
Edu,dvλu,λvEdu,dvλu,λvEdu,dvλu,λvEdu,dvλu,λvEdu,dvλu,λvEdu,dvλu,λvE8,813,148n−2E7,66,916E7,715,168n−2E6,810,98n−2E5,85,118E7,715,154n−3E6,810,1316n−2E5,85,64E6,69,108E6,810,148n−2E7,76,64E6,610,104n−3E7,89,94n−2E7,86,68E8,811,138E7,89,138n−2E7,76,148E8,813,134n−3E7,79,158n−2E7,86,118E8,814,164n−2E8,89,144n−2E6,89,68E8,816,162n−52−1/2+2n−42E8,89,138n−2E6,89,1116E7,816,1616n−22E8,713,1516n−2E8,86,118E8,816,124n−22E7,815,98n−2E8,811,114E7,816,124n−22E7,815,148n−2E8,711,148E7,716,162n−52−1E6,710,98n−2E8,76,144E5,75,68E7,714,158Table 3
Degree-connection number-based partition of vertices ofSKn×n.
VduλuVduλuVduλuVduλuV554V8138n−2V768V7144V864V8144n−2V698V7158n−2V794n−2V7164n−22V894n−2V8164n−22V6108n−2V812n−22V8118


## 2. Connection Number-Based Topological Indices

Letκ=Vκ,Eκ be a graph, where Vκ and Eκ denote the set of vertices and edges accordingly; for a vertex x ∈Vκ, the degree dκx is the number of those vertices which are distance at one from x, and connection number λκx is the number of such vertices which are at distance two from vertex x. For more graph-theoretic notions, see [3, 5, 6].Definition 1 (see [23]).
Letκ be a graph, then the first Zagreb connection index is(1)Z^C1κ=∑v∈Vκλκv2.Definition 2 (see [23]).
Letκ be a graph, then the second Zagreb connection index is(2)Z^C2κ=∑uv∈Eκλκu×λκv,where λκu and λκv represent the connection number of the vertices u and v for the edge uv.Definition 3 (see [23]).
Letκ be a graph, then the modified first Zagreb connection index, modified second Zagreb connection index, modified third Zagreb connection index, and modified fourth Zagreb connection are(i)
aZ^C1∗κ=∑uv∈Eκλκu+λκv(ii)
bZ^C2∗κ=∑uv∈Eκdκuλκv+dκvλκucZ^C3∗κ=∑uv∈Eκdκuλκu+dκvλκudZ^C4∗κ=∑uv∈Eκdκuλκv×dκvλκu=∑uv∈Eκdκuλκu×dκvλκv
whereλκu and λκv represent the connection number of the vertices u and v for the edge uv.Definition 4 (see [23]).
Letκ be a graph, then first multiplicative Zagreb connection index MZ^C1κ, second multiplicative Zagreb connection index MZ^C2κ, third multiplicative Zagreb connection index MZ^C3κ, and fourth multiplicative Zagreb connection index MZ^C4κ are as follows:(i)
aMZ^C1κ=∏v∈Vκλκv2bMZ^C2κ=∏uv∈Eκλκu×λκvcMZ^C3κ=∏uv∈Vκdκu×λκvdMZ^C4κ=∏uv∈Eκλκu+λκvDefinition 5 (see [23]).
For a graphκ, modified first multiplicative Zagreb connection index MZ^C1∗κ, modified second multiplicative Zagreb connection index MZ^C2∗κ, and modified third multiplicative Zagreb connection index MZ^C3∗κ are as follows:aMZ^C1∗κ=∏uv∈Eκdκuλκv+dκvλκubMZ^C2∗κ=∏uv∈Eκdκuλκu+dκvλκvcMZ^C3∗κ=∏uv∈Eκdκuλκv×dκvλκu=∏uv∈Eκdκuλκu×dκvλκv


## 3. Construction of Sudoku Graph

For some integral valuen≥2, Sudoku graph SKn×n is a graphical representation of the Sudoku puzzle in the form of a n×n grid consisting of n2 cells, where each cell is an individual graph of a nodes with four vertices of degree 5, four vertices of degree 7, and one vertex of degree 8. Moreover, for Sudoku graphs κ≅SKn×n, Vκ=9n2 are the number of vertices, and Eκ=n34n−6 are the number of edges, respectively. For n=3, the Sudoku graph SK3×3 is shown in Figure 1.Figure 1
Sudoku graph.SK3×3.In Table1, we computed the results for the connection number-based partition of edges for n≥3 of Sudoku graph SKn×n. In Table 2 and Table 3, we also computed the results for the degree-connection number-based partition of edges and vertices for n≥3 of Sudoku graph SKn×n.


## 4. Connection Number-Based Zagreb Indices

This section deals with the results of the connection number-based Zagreb indices of the Sudoku graphs.Theorem 1.
Letκ≅SKn×n be a Sudoku graph, then its first Zagreb connection index Z^C1κ is(3)Z^C1κ=2192n2−3382n+932.proof.
By Definition1,(4)Z^C1κ=∑v∈Vκλκv2.
We consider the partition of the vertices of the Sudoku graphs with respect to their connection numbers,(5)V1=x∈vκ:λκx=5V2=x∈vκ:λκx=6V3=x∈vκ:λκx=9V4=x∈vκ:λκx=10V5=x∈vκ:λκx=11V6=x∈vκ:λκx=12V7=x∈vκ:λκx=13V8=x∈vκ:λκx=14V9=x∈vκ:λκx=15V10=x∈vκ:λκx=16,such that V1=4, V2=12, V3=8n−1, V4=8n−2, V5=8, V6=n−22, V7=8n−2, V8=4n−1, V9=8n−2, and V10=8n−22.(6)Z^C1κ=∑v∈V1κλκv2+∑v∈V2κλκv2+∑v∈V3κλκv2+∑v∈V4κλκv2+∑v∈V5κλκv2+∑v∈V6κλκv2+∑v∈V7κλκv2+∑v∈V8κλκv2+∑v∈V9κλκv2+∑v∈V10κλκv2Z^C1κ=V152+V262+V392+V4102+V5112+V6122+V7132+V8142+V9152+V10162Z^C1κ=452+1262+8n−192+8n−2102+8112+n−22122+8n−2132+4n−1142+8n−2152+8n−22162.
By performing some calculation, we get(7)Z^C1κ=2192n2−3382n+932.Theorem 2.
Letκ≅SKn×n be a Sudoku graph. Then, its second Zagreb connection index is given as Z^C2κ=8192n2−13756n+4204.Proof.
By Definition2 and Table 1,(8)Z^C2κ=∑uv∈Eκλκu×λκvZ^C2κ=ZC2κ=125×6+169×11+1611×6+126×6+411×11+28n−1613×15+82n−39×10+28n−169×15+4n−89×9+8n−1613×14+8n−2212×16+8n−1615×16+4n−310×10+4n−315×15+246×9+811×14+126×14+811×5+28n−1610×13+8n−114×15+28n−169×13+4n−89×14+8n−210×14+26n2−110n+11616×16+811×13+4n−214×16+4n−313×13.
By performing some calculations, we get(9)Z^C2κ=8192n2−13756n+4204.Theorem 3.
Letκ≅SKn×n be a Sudoku graph. Then, its modified first Zagreb connection index, modified second Zagreb connection index, modified third Zagreb connection index, and the modified fourth Zagreb connection index are as follows:aZ^C1∗κ=1056n2−1164n+180bZ^C2∗κ=8016n2−10388n+2808cZ^C3∗κ=8000n2−8812n+980dZ^C4∗κ=470016n2−853316n+259092Proof.
(a). By Definition3 and Table 1,(10)Z^C1∗κ=∑uv∈Eκλκu+λκvZ^C1∗κ=125+6+169+11+1611+6+126+6+411+11+28n−1613+15+82n−39+10+28n−169+15+4n−89+9+8n−1613+14+8n−2212+16+8n−1615+16+4n−310+10+4n−315+15+246+9+811+14+126+14+811+5+28n−1610+13+8n−114+15+28n−169+13+4n−89+14+8n−210+14+26n2−110n+11616+16+811+13+4n−214+16+4n−313+13.
By performing some calculations, we get(11)Z^C1∗κ=1056n2−1164n+180.
(b). As we know from Definition3,(12)Z^C2∗κ=∑uv∈Eκdκuλκv+dκvλκuZ^C2∗κ=∑uv∈E8,813,14dκuλκv+dκvλκu+∑uv∈E6,810,9dκuλκv+dκvλκu+∑uv∈E6,810,13dκuλκv+dκvλκu+∑uv∈E6,810,14dκuλκv+dκvλκu+∑uv∈E7,89,9dκuλκv+dκvλκu+∑uv∈E7,89,13dκuλκv+dκvλκu+∑uv∈E7,79,15dκuλκv+dκvλκu+∑uv∈E8,89,14dκuλκv+dκvλκu+∑uv∈E8,89,13dκuλκv+dκvλκu+∑uv∈E8,713,15dκuλκv+dκvλκu+∑uv∈E7,815,9dκuλκv+dκvλκu+∑uv∈E7,815,14dκuλκv+dκvλκu+∑uv∈E6,710,9dκuλκv+dκvλκu+∑uv∈E5,75,6dκuλκv+dκvλκu+∑uv∈E7,66,9dκuλκv+dκvλκu+∑uv∈E5,85,11dκuλκv+dκvλκu+∑uv∈E5,85,6dκuλκv+dκvλκu+∑uv∈E7,76,6dκuλκv+dκvλκu+∑uv∈E7,86,6dκuλκv+dκvλκu+∑uv∈E7,76,14dκuλκv+dκvλκu+∑uv∈E7,86,11dκuλκv+dκvλκu+∑uv∈E6,89,6dκuλκv+dκvλκu+∑uv∈E6,89,11dκuλκv+dκvλκu+∑uv∈E8,86,11dκuλκv+dκvλκu+∑uv∈E8,811,11dκuλκv+dκvλκu+∑uv∈E8,711,14dκuλκv+dκvλκu+∑uv∈E8,76,14dκuλκv+dκvλκu+∑uv∈E7,714,15dκuλκv+dκvλκu+∑uv∈E7,715,16dκuλκv+dκvλκu+∑uv∈E7,715,15dκuλκv+dκvλκu+∑uv∈E6,69,10dκuλκv+dκvλκu+∑uv∈E6,610,10dκuλκv+dκvλκu+∑uv∈E8,811,13dκuλκv+dκvλκu+∑uv∈E8,813,13dκuλκv+dκvλκu+∑uv∈E8,814,16dκuλκv+dκvλκu+∑uv∈E7,816,16dκuλκv+dκvλκu+∑uv∈E8,816,12dκuλκv+dκvλκu+∑uv∈E7,816,12dκuλκv+dκvλκu+∑uv∈E7,716,16dκuλκv+dκvλκu+∑uv∈E8,816,16dκuλκv+dκvλκu+E7,79,157×15+7×9+E8,89,148×14+8×9+E8,89,138×13+8×9+E8,713,158×15+7×13+E7,815,97×9+8×15+E7,815,147×14+8×15+E6,710,96×9+7×10+E5,75,65×6+5×7+E7,66,97×9+6×6+E6,89,66×6+8×9+E6,89,116×11+8×9+E8,86,118×11+8×6+E8,811,118×11+8×11+E8,711,148×14+7×11+E8,76,148×14+7×6+E8,813,138×13+8×13+E8,816,148×14+8×16+E7,716,167×16+8×16+E8,816,128×12+8×16+E7,816,127×12+8×16+E7,716,167×16+7×16+E8,816,168×16+8×16.
By using degree-connection number-based partition of edges ofSKn×n as given in Table 2,(13)Z^C2∗κ=8n−2216+8n−2134+16n−2158+8n−2164+4n−2135+8n−2163+8n−2168+4n−2184+8n−2176+16n−2211+8n−2183+8n−2218+8n−2124+865+1699+895+470+484+890+8140+8125+8108+16138+8136+4176+8189+4154++8203+8n−2217+4n−3210+8114+4n−3120+8192×4n−3208+4n−2240+16n−22240+4n−22224+4n−22212+2n−52−1224+2n−52−12+2n−42256.
By performing some calculations, we get(14)Z^C2∗κ=8016n2−10388n+2808.
(c). As we know from Definition3,(15)Z^C3∗κ=∑uv∈Eκdκuλκu+dκvλκvZ^C3∗κ=∑uv∈E8,813,14dκuλκu+dκvλκv+∑uv∈E6,810,9dκuλκu+dκvλκv+∑uv∈E6,810,13dκuλκu+dκvλκv+∑uv∈E6,810,14dκuλκu+dκvλκv+∑uv∈E7,89,9dκuλκu+dκvλκv+∑uv∈E7,89,13dκuλκu+dκvλκv+∑uv∈E7,79,15dκuλκu+dκvλκv+∑uv∈E8,89,14dκuλκu+dκvλκv+∑uv∈E8,89,13dκuλκu+dκvλκv+∑uv∈E8,713,15dκuλκu+dκvλκv+∑uv∈E7,815,9dκuλκu+dκvλκv+∑uv∈E7,815,14dκuλκu+dκvλκv+∑uv∈E6,710,9dκuλκu+dκvλκv+∑uv∈E5,75,6dκuλκu+dκvλκv+∑uv∈E7,66,9dκuλκu+dκvλκv+∑uv∈E5,85,11dκuλκu+dκvλκv+∑uv∈E5,85,6dκuλκv+dκvλκu+∑uv∈E7,76,6dκuλκu+dκvλκv+∑uv∈E7,86,6dκuλκu+dκvλκv+∑uv∈E7,76,14dκuλκu+dκvλκv+∑uv∈E7,86,11dκuλκu+dκvλκv+∑uv∈E6,89,6dκuλκu+dκvλκv+∑uv∈E6,89,11dκuλκu+dκvλκv+∑uv∈E8,86,11dκuλκu+dκvλκv+∑uv∈E8,811,11dκuλκu+dκvλκv+∑uv∈E8,711,14dκuλκu+dκvλκv+∑uv∈E8,76,14dκuλκu+dκvλκv+∑uv∈E7,714,15dκuλκu+dκvλκv+∑uv∈E7,715,16dκuλκu+dκvλκv+∑uv∈E7,715,15dκuλκu+dκvλκv+∑uv∈E6,69,10dκuλκu+dκvλκv+∑uv∈E6,610,10dκuλκu+dκvλκv+∑uv∈E8,811,13dκuλκu+dκvλκv+∑uv∈E8,813,13dκuλκu+dκvλκv+∑uv∈E8,814,16dκuλκu+dκvλκv+∑uv∈E7,816,16dκuλκu+dκvλκv+∑uv∈E8,816,12dκuλκu+dκvλκv+∑uv∈E7,816,12dκuλκu+dκvλκv+∑uv∈E7,716,16dκuλκu+dκvλκv+∑uv∈E8,816,16dκuλκu+dκvλκv.
By using degree-connection number-based partition of edges ofSKn×n as given in Table 2,(16)Z^C3∗κ=8n−2216+8n−2132+16n−2164+8n−2172+4n−2135+8n−2167+8n−2183+4n−2184+8n−2176+16n−2209+8n−2177+8n−2217+8n−2123+867+1696+8113+473+484+890+8140+8130+8102+16142+8136+4176+8186+4146+8203+8n−2217+4n−3210+8114+4n−3120+8192+4n−3208+4n−2240+16n−22240+4n−22224+4n−22208+2242n−52−1+2n−52−12+2n−42256.
By performing some calculations, we get(17)Z^C3∗κ=8000n2−8812n+980.
(d). As we know from Definition3,(18)Z^C4∗κ=∑uv∈Eκdκuλκu×dκvλκvZ^C4∗κ=∑uv∈E8,813,14dκuλκu×dκvλκv+∑uv∈E6,810,9dκuλκu×dκvλκv+∑uv∈E6,810,13dκuλκu×dκvλκv+∑uv∈E6,810,14dκuλκu×dκvλκv+∑uv∈E7,89,9dκuλκu×dκvλκv+∑uv∈E7,89,13dκuλκu×dκvλκv+∑uv∈E7,79,15dκuλκu×dκvλκv+∑uv∈E8,89,14dκuλκu×dκvλκv+∑uv∈E8,89,13dκuλκu×dκvλκv+∑uv∈E8,713,15dκuλκu×dκvλκv+∑uv∈E7,815,9dκuλκu×dκvλκv+∑uv∈E7,815,14dκuλκu×dκvλκv+∑uv∈E6,710,9dκuλκu×dκvλκv+∑uv∈E5,75,6dκuλκu×dκvλκv+∑uv∈E7,66,9dκuλκu×dκvλκv+∑uv∈E5,85,11dκuλκu×dκvλκv+∑uv∈E5,85,6dκuλκv×dκvλκu+∑uv∈E7,76,6dκuλκu×dκvλκv+∑uv∈E7,86,6dκuλκu×dκvλκv+∑uv∈E7,76,14dκuλκu×dκvλκv+∑uv∈E7,86,11dκuλκu×dκvλκv+∑uv∈E6,89,6dκuλκu×dκvλκv+∑uv∈E7,86,6dκuλκu×dκvλκv+∑uv∈E7,76,14dκuλκu×dκvλκv+∑uv∈E7,86,11dκuλκu×dκvλκv+∑uv∈E6,89,6dκuλκu×dκvλκv+∑uv∈E6,89,11dκuλκu×dκvλκv+∑uv∈E8,86,11dκuλκu×dκvλκv+∑uv∈E8,811,11dκuλκu×dκvλκv+∑uv∈E8,711,14dκuλκu×dκvλκv+∑uv∈E8,76,14dκuλκu×dκvλκv+∑uv∈E7,714,15dκuλκu×dκvλκv+∑uv∈E7,715,16dκuλκu×dκvλκv+∑uv∈E7,715,15dκuλκu×dκvλκv+∑uv∈E8,76,14dκuλκu×dκvλκv+∑uv∈E7,714,15dκuλκu×dκvλκv+∑uv∈E7,715,16dκuλκu×dκvλκv+∑uv∈E7,715,15dκuλκu×dκvλκv+∑uv∈E8,76,14dκuλκu×dκvλκv+∑uv∈E7,714,15dκuλκu×dκvλκv+∑uv∈E7,715,16dκuλκu×dκvλκv+∑uv∈E7,715,15dκuλκu×dκvλκv+∑uv∈E6,69,10dκuλκu×dκvλκv+∑uv∈E6,610,10dκuλκu×dκvλκv+∑uv∈E8,811,13dκuλκu×dκvλκv+∑uv∈E8,813,13dκuλκu×dκvλκv+∑uv∈E8,814,16dκuλκu×dκvλκv+∑uv∈E7,816,16dκuλκu×dκvλκv+∑uv∈E8,816,12dκuλκu×dκvλκv+∑uv∈E7,816,12dκuλκu×dκvλκv+∑uv∈E7,716,16dκuλκu×dκvλκv+∑uv∈E8,816,16dκuλκu×dκvλκv.
By using degree-connection number-based partition of edges ofSKn×n as given in Table 2,(19)Z^C4∗κ=8n−211648+8n−24320+16n−26240+8n−26720+4n−24536+8n−26552+8n−26615+4n−28064+8n−27488+16n−210920+8n−27560+8n−211760+8n−23780+81050+162268+82200+41200+41764+82016+84116+83696+82592+164752+84224+47744+82464+44704+810290+8n−211760+4n−311025+83240+4n−33600+89152+4n−310816+4n−214336+16n−2214336+4n−2212288+4n−2210752+2n−52−112544+2n−52−12−2n−4216384.By performing some calculations, we get(20)Z^C4∗κ=470016n2−853316n+259092.


## 5. Connection Number-Based Multiplicative Zagreb Indices

This section deals with the results of the connection number-based multiplicative Zagreb indices of the Sudoku graphs.Theorem 4.
Letκ≅SKn×n be a Sudoku graph. Then, its first multiplicative Zagreb connection index, second multiplicative Zagreb connection index, third multiplicative Zagreb connection index, and fourth multiplicative Zagreb connection index are as follows:aMZ^C1κ=268n2−248n+256×32n2+40n−32×532n−56×78n−1×1116×1316n−2bMZ^C2κ=2256n2−944n+980×38n2+144n−136×5104n−188×732n−36×1164×1364n−2cMZ^C3κ=249n2−112n+104×3n2+28n−24×516n−24×74n2×118×138n−2dMZ^C4κ=2146n2−470n+456×364n−96×512n+3×78n2−48n+32×1116n−1×134n−3.×1716×1982n−3×2320n−2×298n−1×318n−2Proof.
(a) By Definition4,(21)MZ^C1κ=∏v∈Vκλκv2MZ^C1κ=∏v∈V1κλκv2×∏v∈V2κλκv2×∏v∈V3κλκv2×∏v∈V4κλκv2×∏v∈V5κ×∏v∈V6κλκv2×∏v∈V7κλκv2×∏v∈V8κλκv2×∏v∈V9κλκv2×∏v∈V10κλκv2.
By using the partition of the vertices of the Sudoku graphs with respect to their connection numbers, given in Theorem1, we get(22)MZ^C1κ=52V1×62V2×92V3×102V4×112V5×122V6×132V7×142V8×152V9×162V10MZ^C1κ=52×4×62×12×92×8n−1×102×8n−2×112×8×122×n−22×132×8n−2×142×4n−1×152×8n−2+×162×8n−22.
By performing some calculations, we getMZ^C1κ=268n2−248n+256×32n2+40n−32×532n−56×78n−1×1116×1316n−2.
(b) As we know from Definition4,(23)MZ^C2κ=∏uv∈Eκλκu×λκvMZ^C2κ=∏uv∈E5,6cλκu×λκv×∏uv∈E9,11cλκu×λκv×∏uv∈E11,6cλκu×λκv×∏uv∈E6,6cλκu×λκv×∏uv∈E11,11cλκu×λκv×∏uv∈E13,15cλκu×λκv×∏uv∈E9,10cλκu×λκv×∏uv∈E9,15cλκu×λκv×∏uv∈E9,9cλκu×λκv×∏uv∈E13,14cλκu×λκv×∏uv∈E12,16cλκu×λκv×∏uv∈E15,16cλκu×λκv×∏uv∈E10,10cλκu×λκv×∏uv∈E15,15cλκu×λκv×∏uv∈E6,9cλκu×λκv×∏uv∈E11,14cλκu×λκv×∏uv∈E6,14cλκu×λκv×∏uv∈E11,5cλκu×λκv×∏uv∈E10,13cλκu×λκv×∏uv∈E14,15cλκu×λκv×∏uv∈E9,13cλκu×λκv×∏uv∈E9,14cλκu×λκv×∏uv∈E10,14cλκu×λκv×∏uv∈E16,16λκu×λκv×∏uv∈E11,13cλκu×λκv×∏uv∈E14,16cλκu×λκv×∏uv∈E13,13cλκu×λκv.
By using connection-based partition of edges ofSKn×n given in Table 1,(24)MZ^C2κ=5×612×9×1116×11×616×6×612×11×114×13×1528n−16×9×1082n−3×9×1528n−16×9×94n−8×13×148n−16×12×168n−22×15×168n−16×10×104n−3×15×154n−3×6×924×11×148×6×1412×11×58×10×1328n−16×14×158n−1×9×1328n−16×9×144n−8×10×148n−2×16×1626n2−110n+116×11×138×14×164n−2×13×134n−3.
By performing some calculations, we get(25)MZ^C2κ=2256n2−944n+980×38n2+144n−136×5104n−188×732n−36×1164×1364n−2.
(c) As we know from Definition4,(26)MZ^C3κ=∏v∈Vκdκv×λκvMZ^C3κ=∏v∈V1κdκv×λκv×∏v∈V2κdκv×λκv×∏v∈V3κdκv×λκv×∏v∈V4κdκu×λκv×∏v∈V5κdκv×λκv×∏v∈V6κdκv×λκv×∏v∈V7κdκv×λκv×∏v∈V8κdκv×λκv×∏v∈V9κdκv×λκv×∏v∈V10κdκv×λκv×∏v∈V11κdκv×λκv×∏v∈V12κdκv×λκv×∏v∈V13κdκu×λκv×∏v∈V14κdκv×λκv×∏v∈V15κdκv×λκv.By using degree-connection number-based partition of vertices of SKn×n as given in Table 3,(27)MZ^C3κ=5×54×6×78×6×84×9×68×9×74n−2×9×84n−2×10×68n−2×11×88×13×88n−2×14×74×14×84n−2×15×78n−2×16×74n−22×16×84n−22×12×8n−22.
By performing some calculations, we get(28)MZ^C3κ=249n2−112n+104×3n2+28n−24×516n−24×74n2×118×138n−2.
(d) As we know from Definition4,(29)MZ^C4κ=∏uv∈E5,6cλκu+λκv×∏uv∈E9,11cλκu+λκv×∏uv∈E11,6cλκu+λκv×∏uv∈E6,6cλκu+λκv×∏uv∈E11,11cλκu+λκv×∏uv∈E13,15cλκu+λκv×∏uv∈E9,10cλκu+λκv×∏uv∈E9,15cλκu+λκv×∏uv∈E9,9cλκu+λκv×∏uv∈E13,14cλκu+λκv×∏uv∈E12,16cλκu+λκv×∏uv∈E15,16cλκu+λκv×∏uv∈E10,10cλκu+λκv×∏uv∈E15,15cλκu+λκv×∏uv∈E6,9cλκu+λκv×∏uv∈E11,14cλκu+λκv×∏uv∈E6,14cλκu+λκv×∏uv∈E11,5cλκu+λκv×∏uv∈E10,13cλκu+λκv×∏uv∈E14,15cλκu+λκv×∏uv∈E9,13cλκu×λκv×∏uv∈E9,14cλκu+λκv×∏uv∈E10,14cλκu+λκv×∏uv∈E16,16cλκu+λκv×∏uv∈E11,13cλκu+λκv×∏uv∈E14,16cλκu+λκv×∏uv∈E13,13cλκu+λκv.By using connection number-based partition of edges of SKn×n as given in Table 1,(30)MZ^C4κ=5+612×9+1116×11+616×6+612×11+114×13+1528n−16×9+1082n−3×9+1528n−16×9+94n−8×13+148n−16×12+168n−22×15+168n−16×10+104n−3×15+154n−3×6+924×11+148×6+1412×11+58×10+1328n−16×14+158n−1×9+1328n−16×9+144n−8×10+148n−2×16+1626n2−110n+116×11+138×14+164n−2×13+134n−3.
By performing some calculations, we get(31)MZ^C4κ=2146n2−470p+456×364n−96×512n+3×78n2−48n+32×1116n−1×134n−3×1716×1982n−3+×2320n−2×298n−1×318n−2.Theorem 5.
Letκ≅SKn×n be a Sudoku graph. Then, the modified first multiplicative Zagreb connection index, modified second multiplicative Zagreb connection index, and modified third multiplicative Zagreb connection index are as follows:aMZ^C1∗κ=2160n2−472n+440×316n2+60×516n2−48n+80×78n2−16n+32×118n+1×134n−1×178×1916×234n+2×298×3116n−2×418n−2×534n−22×618n−2×678n−2×7916n−32×1098n−16×1638n−2×21116n−2,bMZ^C2∗κ=2168n2−504n+556×316n2+16n−52×516n2−48n+48×78n2−16n+16×1132n−60×134n2−12n+12×178×1916n−2×234n−2×298×3116n−24×4124n−2×438n−2×518×578×598n−2×678×7116×7312×1138×1678n−2,cMZ^C3∗κ=2376n2−992n+932×38n2+192n−184×5104n−168×728n2+4n−8×1164×1364n−2.
Proof (a) By Definition5,(32)MZ^C1∗κ=∏uv∈Eκdκuλκv+dκvλκuMZ^C1∗κ=∏uv∈E8,813,14dκuλκv+dκvλκu×∏uv∈E6,810,9dκuλκv+dκvλκu×∏uv∈E6,810,13dκuλκv+dκvλκu×∏uv∈E6,810,14dκuλκv+dκvλκu×∏uv∈E7,89,9dκuλκv+dκvλκu×∏uv∈E7,89,13dκuλκv+dκvλκu×∏uv∈E7,79,15dκuλκv+dκvλκu×∏uv∈E8,89,14dκuλκv+dκvλκu×∏uv∈E8,89,13dκuλκv+dκvλκu∏uv∈E8,713,15dκuλκv+dκvλκu×∏uv∈E7,815,9dκuλκv+dκvλκu×∏uv∈E7,815,14dκuλκv+dκvλκu×∏uv∈E6,710,9dκuλκv+dκvλκu×∏uv∈E5,75,6dκuλκv+dκvλκu×∏uv∈E7,66,9dκuλκv+dκvλκu×∏uv∈E5,85,11dκuλκv+dκvλκu×∏uv∈E5,85,6dκuλκv+dκvλκu×∏uv∈E7,76,6dκuλκv+dκvλκu×∏uv∈E7,86,6dκuλκv+dκvλκu×∏uv∈E7,76,14dκuλκv+dκvλκu×∏uv∈E7,86,11dκuλκv+dκvλκu×∏uv∈E6,89,6dκuλκv+dκvλκu×∏uv∈E6,89,11dκuλκv+dκvλκu×∏uv∈E8,86,11dκuλκv+dκvλκu×∏uv∈E8,811,11dκuλκv+dκvλκu×∏uv∈E8,711,14dκuλκv+dκvλκu×∏uv∈E8,76,14dκuλκv+dκvλκu×∏uv∈E7,714,15dκuλκv+dκvλκu×∏uv∈E7,715,16dκuλκv+dκvλκu×∏uv∈E7,715,15dκuλκv+dκvλκu×∏uv∈E6,69,10dκuλκv+dκvλκu×∏uv∈E6,610,10dκuλκv+dκvλκu×∏uv∈E8,811,13dκuλκv+dκvλκu×∏uv∈E8,813,13dκuλκv+dκvλκu×∏uv∈E8,814,16dκuλκv+dκvλκu×∏uv∈E7,816,16dκuλκv+dκvλκu×∏uv∈E6,69,10dκuλκv+dκvλκu×∏uv∈E6,610,10dκuλκv+dκvλκu×∏uv∈E8,811,13dκuλκv+dκvλκu×∏uv∈E8,813,13dκuλκv+dκvλκu×∏uv∈E8,814,16dκuλκv+dκvλκu×∏uv∈E7,816,16dκuλκv+dκvλκu×∏uv∈E8,816,12dκuλκv+dκvλκu×∏uv∈E7,816,12dκuλκv+dκvλκu×∏uv∈E7,716,16dκuλκv+dκvλκu×∏uv∈E8,816,16dκuλκv+dκvλκu.
By using degree-connection number-based partition of edges ofSKn×n as given in Table 2, we get(33)MZ^C1∗κ=2168n−2×1348n−2×15816n−2×1648n−2×1354n−2×1638n−2×1688n−2×1844n−2×1768n−2×21116n−2×1838n−2×2188n−2×1248n−2×658×9916×958×704×844×908×1408×1258×1088×13816×1368×1764×1898×1544××2038×2178n−2×2104n−3×1548×1204n−3×1928×2084n−3×2404n−2×24016n−22×2244n−22×2124n−22×2242n−52−1×2562n−52−12−2n−42.
By performing some calculations, we get(34)MZ^C1∗κ=2160n2−472n+440×316n2+60×516n2−48n+80×78n2−16n+32×118n+1×134n−1×178×1916×234n+2×298×3116n−2×418n−2×534n−22×618n−2×678n−2×7916n−32×1098n−16×1638n−2×21116n−2.
(b) As we know from Definition5,(35)MZ^C2∗κ=∏uv∈Eκdκuλκu+dκvλκvMZ^C2∗κ=∏uv∈E8,813,14dκuλκu+dκvλκv×∏uv∈E6,810,9dκuλκu+dκvλκv×∏uv∈E6,810,13dκuλκu+dκvλκv×∏uv∈E6,810,14dκuλκu+dκvλκv×∏uv∈E7,89,9dκuλκu+dκvλκv×∏uv∈E7,89,13dκuλκu+dκvλκv×∏uv∈E7,79,15dκuλκu+dκvλκv×∏uv∈E8,89,14dκuλκu+dκvλκv×∏uv∈E8,89,13dκuλκu+dκvλκv×∏uv∈E8,713,15dκuλκu+dκvλκv×∏uv∈E7,815,9dκuλκu+dκvλκv×∏uv∈E7,815,14dκuλκu+dκvλκv×∏uv∈E6,710,9dκuλκu+dκvλκv×∏uv∈E5,75,6dκuλκu+dκvλκv×∏uv∈E7,66,9dκuλκu+dκvλκv×∏uv∈E5,85,11dκuλκu+dκvλκv×∏uv∈E5,85,6dκuλκv+dκvλκu×∏uv∈E7,76,6dκuλκu+dκvλκv×∏uv∈E6,710,9dκuλκu+dκvλκv×∏uv∈E5,75,6dκuλκu+dκvλκv×∏uv∈E7,66,9dκuλκu+dκvλκv×∏uv∈E5,85,11dκuλκu+dκvλκv×∏uv∈E5,85,6dκuλκv+dκvλκu×∏uv∈E7,76,6dκuλκu+dκvλκv×∏uv∈E7,86,6dκuλκu+dκvλκv×∏uv∈E7,76,14dκuλκu+dκvλκv×∏uv∈E7,86,11dκuλκu+dκvλκv×∏uv∈E6,89,6dκuλκu+dκvλκv×∏uv∈E6,89,11dκuλκu+dκvλκv×∏uv∈E8,86,11dκuλκu+dκvλκv×∏uv∈E8,811,11dκuλκu+dκvλκv×∏uv∈E8,711,14dκuλκu+dκvλκv×∏uv∈E8,76,14dκuλκu+dκvλκv×∏uv∈E7,714,15dκuλκu+dκvλκv×∏uv∈E7,715,16dκuλκu+dκvλκv×∏uv∈E7,715,15dκuλκu+dκvλκv×∏uv∈E6,69,10dκuλκu+dκvλκv×∏uv∈E6,610,10dκuλκu+dκvλκv×∏uv∈E8,811,13dκuλκu+dκvλκv×∏uv∈E8,813,13dκuλκu+dκvλκv×∏uv∈E8,814,16dκuλκu+dκvλκv×∏uv∈E7,816,16dκuλκu+dκvλκv×∏uv∈E8,816,12dκuλκu+dκvλκv×∏uv∈E7,816,12dκuλκu+dκvλκv×∏uv∈E7,716,16dκuλκu+dκvλκv×∏uv∈E8,816,16dκuλκu+dκvλκv.
By using degree-connection number-based partition of edges ofSKn×n as given in Table 2,(36)MZ^C2∗κ=2168n−2×1328n−2×16416n−2×1728n−2×1354n−2×1678n−2×1688n−2×1844n−2×1768n−2×20916n−2×1778n−2×2178n−2×1238n−2×678×9616×1138×734×844×908×1408×1308×1028×14216×1368×1764×1868×1464××2038×2178n−2×2104n−3×1148×1204n−3×1928×2084n−3×2404n−2×24016n−22×2244n−22×2084n−22×2242n−52−1×2562n−52−12−2n−42.
By performing some calculations, we get(37)MZ^C2∗κ=2168n2−504n+556×316n2+16n−52×516n2−48n+48×78n2−16n+16×1132n−60×134n2−12n+12×178×1916n−2×234n−2×298×3116n−24×4124n−2×438n−2×518×578×598n−2×678×7116×7312×1138×1678n−2.
(c) As we know from Definition5,(38)MZ^C3∗κ=∏uv∈Eκdκuλκu×dκvλκvMZ^C3∗κ=∏uv∈E8,813,14dκuλκu×dκvλκv×∏uv∈E6,810,9dκuλκu×dκvλκv×∏uv∈E6,810,13dκuλκu×dκvλκv×∏uv∈E6,810,14dκuλκu×dκvλκv×∏uv∈E7,89,9dκuλκu×dκvλκv×∏uv∈E7,89,13dκuλκu×dκvλκv×∏uv∈E7,79,15dκuλκu×dκvλκv×∏uv∈E8,89,14dκuλκu×dκvλκv×∏uv∈E8,89,13dκuλκu×dκvλκv×∏uv∈E8,713,15dκuλκu×dκvλκv×∏uv∈E7,815,9dκuλκu×dκvλκv×∏uv∈E7,815,14dκuλκu×dκvλκv×∏uv∈E6,710,9dκuλκu×dκvλκv×∏uv∈E5,75,6dκuλκu×dκvλκv×∏uv∈E7,66,9dκuλκu×dκvλκv×∏uv∈E5,85,11dκuλκu×dκvλκv×∏uv∈E5,85,6dκuλκv×dκvλκu×∏uv∈E7,76,6dκuλκu×dκvλκvMZ^C3∗κ=∏uv∈E8,813,14dκuλκu×dκvλκv×∏uv∈E6,810,9dκuλκu×dκvλκv×∏uv∈E6,810,13dκuλκu×dκvλκv×∏uv∈E6,810,14dκuλκu×dκvλκv×∏uv∈E7,89,9dκuλκu×dκvλκv×∏uv∈E7,89,13dκuλκu×dκvλκv×∏uv∈E7,79,15dκuλκu×dκvλκv×∏uv∈E8,89,14dκuλκu×dκvλκv×∏uv∈E8,89,13dκuλκu×dκvλκv×∏uv∈E8,713,15dκuλκu×dκvλκv×∏uv∈E7,815,9dκuλκu×dκvλκv×∏uv∈E7,815,14dκuλκu×dκvλκv×∏uv∈E6,710,9dκuλκu×dκvλκv×∏uv∈E5,75,6dκuλκu×dκvλκv×∏uv∈E7,66,9dκuλκu×dκvλκv×∏uv∈E5,85,11dκuλκu×dκvλκv×∏uv∈E5,85,6dκuλκv×dκvλκu×∏uv∈E7,76,6dκuλκu×dκvλκv×∏uv∈E7,86,6dκuλκu×dκvλκv×∏uv∈E7,76,14dκuλκu×dκvλκv×∏uv∈E7,86,11dκuλκu×dκvλκv×∏uv∈E6,89,6dκuλκu×dκvλκv×∏uv∈E6,89,11dκuλκu×dκvλκv×∏uv∈E8,86,11dκuλκu×dκvλκv×∏uv∈E8,811,11dκuλκu×dκvλκv×∏uv∈E8,711,14dκuλκu×dκvλκv×∏uv∈E8,76,14dκuλκu×dκvλκv×∏uv∈E7,714,15dκuλκu×dκvλκv×∏uv∈E7,715,16dκuλκu×dκvλκv×∏uv∈E7,715,15dκuλκu×dκvλκv×∏uv∈E6,69,10dκuλκu×dκvλκv×∏uv∈E6,610,10dκuλκu×dκvλκv×∏uv∈E8,811,13dκuλκu×dκvλκv×∏uv∈E8,813,13dκuλκu×dκvλκv×∏uv∈E8,814,16dκuλκu×dκvλκv×∏uv∈E7,816,16dκuλκu×dκvλκv×∏uv∈E8,816,12dκuλκu×dκvλκv×∏uv∈E7,816,12dκuλκu×dκvλκv×∏uv∈E7,716,16dκuλκu×dκvλκv×∏uv∈E8,816,16dκuλκu×dκvλκv.
By using degree-connection number based partition of edges ofSKn×n as given in Table 2,(39)MZ^C3∗κ=116488n−2×43208n−2×624016n−2×67208n−2×45364n−2×65528n−2×66158n−2×80644n−2×74888n−2×1092016n−2×75608n−2×117608n−2×37808n−2×10508×226816×22008×12004×17644×20168×41168×36968×25928×475216×42248×77444×86248×47044××102908×117608n−2×110254n−3×32408×36004n−3×91528×108164n−3×143364n−2×1433616n−22×122884n−22×107524n−22×125442n−52−1×163842n−52−12−2n−42.By performing some calculations, we get(40)MZ^C3∗κ=2376n2−992n+932×38n2+192n−184×5104n−168×728n2+4n−8×1164×1364n−2.


## 6. Conclusion

In this paper, we constructed the partitions of the edges of the Sudoku graphs with respect to their connection numbers (see Table1) and degree-connection numbers (see Table 2) of their end vertices. Moreover, the partition of the vertex set of the Sudoku graph is obtained with respect to the degree and connection number considering at the same time (see Table 3). Finally, we obtained first and second Zagreb connection, modified first, second, third, and fourth Zagreb connection, first, second, third, and fourth multiplicative Zagreb connection, and modified first, second, and third multiplicative Zagreb connection indices that are computed for Sudoku graphs.



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*Source: 4279702-2022-04-28.xml*