# Chromatic Schultz and Gutman Polynomials of Jahangir GraphsJ2,m and J3,m

Authors: Ramy Shaheen; Suhail Mahfud; Qays Alhawat
Journal: Journal of Applied Mathematics
 (2023)

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## Abstract

Topological polynomial and indices based on the distance between the vertices of a connected graph are widely used in the chemistry to establish relation between the structure and the properties of molecules. In a similar way, chromatic versions of certain topological indices and the related polynomial have also been discussed in the recent literature. In this paper, we present the chromatic Schultz and Gutman polynomials and the expanded form of the Hosoya polynomial and chromatic Schultz and Gutman polynomials, and then we derive these polynomials for special cases of Jahangir graphs.


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## Body

## 1. Introduction

Suppose thatGV,E is a connected undirected graph with vertices set V and the edge set E. Then the distance between two vertices u,v is denoted by du,v, which is defined as the length of shortest path between u and v. The degree of vertex u in G is denoted by du, which is defined as the number of edges incident to u. The diameter of G is denoted by DG, which is defined as DG=Maxu∈Vdu,v:v∈V.Having a molecule, if we represent atoms by vertices and bonds by edges, we obtain a molecular graph [1, 2]. Algebraic polynomial plays a significant part in chemistry because of its important applications for molecular compounds.Topological indices in biology and chemistry were used for the first time in 1947 when chemist Wiener [3] introduced Wiener’s index which is the oldest topological index studied to demonstrate correlations between physicochemical properties of organic compounds of molecular graphs. At first, Wiener’s index was used to predict the boiling of paraffin [4]. Recently, there is a proposal to use Wiener’s index to predict conformational switching in RNA structures [5].In chemistry, drug discovery commonly relies on the topological indices. If we compute topological indices of drug molecular structures, medical and pharmaceutical researchers will be able to understand their therapeutic properties which can compensate for the shortcomings of medicine and chemical experiments.The Hosoya polynomial [6] is one of the known examples which determines distance-based topological indices, and through it, we can obtain the Wiener index. M-polynomial [7] introduced in 2015 plays the same role in determining closed forms of many degree-based topological indices [8, 9]. In [10], we see M-polynomial and some indices of Jahangir graph Jn,m. In [11], we see M-polynomial and some indices of polyhex nanotubes.The Hosoya polynomial of graph is defined asHG,x=1/2∑u∈V∑v∈Vxdu,v was introduced by Haruo Hosoya in 1988 as a counting polynomial. It actually counts the number of distances of path of different lengths in a molecular graph [12]. The Hosoya polynomial is very well studied. In 1993, Gutman introduced the Hosoya polynomial for a vertex of a graph, and these polynomials are correlated. The most interesting application of the Hosoya polynomial is the almost all distance-based graph invariants, which are used to predict physical, chemical, and pharmacological properties of organic molecules, which can be recovered from the Hosoya polynomial [13].The Schultz index of graphG was introduced by Schultz [14] in 1989 as follows:
(1)ScG=12∑u∈V∑v∈Vdu+dvdu,v.The Gutman index of graphG was introduced by Klavẑar and Gutman [15] in 1996 as follows:
(2)GutG=12∑u∈V∑v∈Vdu×dvdu,v.The recent update in the computational techniques of the Schultz and Gutman indices can be found in [16, 17] and implemented with relativistic parameters in [18, 19].Also, the Schultz and Gutman polynomials ofG are defined as ScG,x=1/2∑u∈V∑v∈Vdu+dvxdu,v and GutG,x=1/2∑u∈V∑v∈Vdu+dvxdu,v, respectively, and the relationship between them as follows: GutG=∂/∂xGutG,xx=1,ScG=∂/∂xScG,xx=1 [20].A properk− coloring, or simply k− coloring of graph GV,E is a function φ:V⟶C;C=c1,c2,c3,⋯,ck such that for each uv∈E; φu≠φv. A graph G is k− coloring if there exists k− coloring of G. The chromatic number χG of graph G is the smallest k such that G is C.An independent set [21] is a subset of vertices S⊆V such that no two vertices in S are adjacent. A legal k− coloring of G corresponds to partition of V in to k independent set. We can also define a function ξ:V⟶1,2,3,⋯,k such that ξvi=s if φvi=cs;cs∈C.In order to getφ+ (maximal parameter coloring), we give the largest independent set, color ck, then any vertex in this set will be ξφ+u=k, and if we give the second largest independent set, color ck−1, then any vertex in this set will be ξφ+u=k−1. So, on till give the smallest independent set color c1 then any vertex in this set will be ξφ+u=1.In order to getφ− (minimal parameter coloring), we give the largest independent set, color c1, then any vertex in this set will be ξφ−u=1, and if we give the second largest independent set, color c2, then any vertex in this set will be ξφ−u=2. So, to give the smallest independent set, color ck, then any vertex in this set will be ξφ−u=k.Definition 1 (See [22]).
LetG be a connected graph with chromatic number χG. Then the chromatic Schultz polynomial of G denoted by SχG,x is defined as
(3)SχG,x=12∑u∈V∑v∈Vξu+ξvxdu,v.Definition 2 (See [22]).
LetG be a connected graph with chromatic number χG. Then the chromatic Gutman polynomial of G denoted by GutχG,x is defined as
(4)GutχG,x=12∑u∈V∑v∈Vξu×ξvxdu,v.Definition 3 (See [22]).
LetG be a connected graph with chromatic number φ− and φ+ be the minimal and maximal parameter coloring of G. Then,
(1)
Theχ+− chromatic Schultz polynomial of G, denoted by Sχ+, is defined as Sχ+G,x=1/2∑u∈V∑v∈Vξφ+u+ξφ+vxdu,v(2)
Theχ+− chromatic Gutman polynomial of G, denoted by Gutχ+, is defined as Gutχ+G,x=1/2∑u∈V∑v∈Vξφ+u×ξφ+vxdu,v(3)
Theχ−− chromatic Schultz polynomial of G, denoted by Sχ−, is defined as: Sχ−G,x=1/2∑u∈V∑v∈Vξφ−u+ξφ−vxdu,v(4)
Theχ−− chromatic Gutman polynomial of G, denoted by Gutχ−, is defined as Gutχ−G,x=1/2∑u∈V∑v∈Vξφ−u×ξφ−vxdu,v
Note: when calculating the Hosoya polynomial, we do not calculate the distance between the vertex and itself. When calculating a chromatic Schultz and Gutman polynomials, we calculate the distance of the vertex from itself.The Hosoya polynomial gives only the number and lengths of paths between the vertices of the graph. If we have the termnxm in the Hosoya polynomial, this will indicate that n is the number of paths of length m between the vertices in graph G. In this search, we will suggest new definitions which will be generalization for the Hosoya polynomial. The definitions will limit, in addition to the number and lengths of paths between the vertices of the graph, the colors of vertices in the beginning and the end of every path. As a result, the Hosoya polynomial becomes a special case from the suggested definitions. So, we can find the Hosoya polynomial from the new definitions, and these definitions areDefinition 4.
LetG be a connected graph. Then, the expanded H+− Hosoya polynomial denoted by H+G,x,k is defined as
(5)H+G,x,k=12∑i=1k∑u∈Vi+∑v∈Vi+xidu,v+∑i=1k∑j=1:j≠ik∑u∈Vi+∑v∈Vj+xi,jdu,v,where Vi+=v∈V:ξφ+v=i;1≤i≤k,Vj+=v∈V:ξφ+v=j;1≤j≤k,HG,x=H+G,x,kxi=xxi,j=x.Definition 5.
LetG be a connected graph. Then, the expanded H−− Hosoya polynomial denoted by H−G,x,k is defined as
(6)H−G,x,k=12∑i=1k∑u∈Vi−∑v∈Vi−xidu,v+∑i=1k∑j=1:j≠ik∑u∈Vi−∑v∈Vj−xi,jdu,vHG,x=H−G,x,kxi=xxi,j=x.In order to clarify these suggested new definitions, to determine the difference between it and the Hosoya polynomial, and to know the new things it adds, we will present the following example. In Figure1, we have the graph P5.Figure 1
The graphP5.We will compute the Hosoya polynomial forP5.
(7)HP5,x=12∑u∈V∑v∈Vxdu,v=x4+2x3+3x2+4x1.By the Hosoya polynomial ofP5, we can find out the number and length of the paths between the vertices of P5. The paths are as follows:Four paths of length one, three paths of length two, two paths of length three, and one paths of length four.The pathP5 has two independent sets v1,v3,v5 and v2,v4, thus χP5=2. In order to compute expanded H+− Hosoya polynomial, we will give the vertices of set v1,v3,v5 the color c2 and the vertices of set v2,v4 the color c1, so we have φ:V⟶c1,c2:φv1=c2,φv2=c1,φv3=c2,φv4=c1,φv5=c2. We also, have ξ:V⟶1,2:ξφ+v1=2, ξφ+v2=1,ξφ+v3=2,ξφ+v4=1,ξφ+v5=2.(8)V1+=v2,v4,V2+=v1,v3,v5,H+P5,x,2=12∑i=12∑u∈Vi+∑v∈Vi+xidu,v+∑i=12∑j=1:j≠i2∑u∈Vi+∑v∈Vj+xi,jdu,v=x24+2x1,23+x12+2x22+4x1,21.By expandedH+− Hosoya polynomial of P5, we can find that there is one path of length four has this path starting and ending with vertex having the color c2, and two paths of length three where each path starting with vertex having the color c1 and ending with vertex having the color c2. One path of length two has this path starting and ending with vertex having the color c1, two paths of length two have each path starting and ending with vertex have the color c2, and four paths of length one each path starting with vertex having the color c1 and ending with vertex has the color c2.
(9)HP5,x=H+P5,x,2x1=xx2=xx1,2=x=x4+2x3+3x2+4x1.To compute expandedH−− Hosoya polynomial of P5, we will give the vertices of set v1,v3,v5 the color c1 and the vertices of set v2,v4 the color c2 so we have
(10)φ:V⟶c1,c2:φv1=c1,φv2=c2,φv3=c1,φv4=c2,φv5=c1ξ:V⟶1,2:ξφ−v1=1,ξφ−v2=2,ξφ−v3=1,ξφ−v4=2,ξv−v5=1,V1−=v1,v3,v5,V2−=v2,v4,H−P5,x,2=12∑i=12∑u∈Vi−∑v∈Vi−xidu,v+∑i=12∑j=1:j≠i2∑u∈Vi−∑v∈Vj−xi,jdu,v=x14+2x1,23+2x12+x22+4x1,21,HP5,x=H−P5,x,2x1=xx2=xx1,2=x=x4+2x3+3x2+4x1.In this search we also suggest new definitions which will be generalization for the chromatic Schultz and Gutman polynomials where the new definitions will limit the colors of vertices in the beginning and the end of every path in the graph and these definitions areDefinition 6.
LetG be a connected graph. Then, the expanded χ+− chromatic Schultz polynomial of G, denoted by Sχ+G,x,k, is defined as
(11)Sχ+G,x,k=12∑i=1k∑u∈Vi+∑v∈Vi+ξφ+u+ξφ+vxidu,v+∑i=1k∑j=1:j≠ik∑u∈Vi+∑v∈Vj+ξφ+u+ξφ+vxi,jdu,v,Sχ+G,x=Sχ+G,x,kxi=xxi,j=x.Definition 7.
LetG be a connected graph. Then, the expanded χ+− chromatic Gutman polynomial of G, denoted by Gutχ+G,x,k, is defined as
(12)Gutχ+G,x,k=12∑i=1k∑u∈Vi+∑v∈Vi+ξφ+u×ξφ+vxidu,v+∑i=1k∑j=1:j≠ik∑u∈Vi+∑v∈Vj+ξφ+u×ξφ+vxi,jdu,vGutχ+G,x=Gutχ+G,x,kxi=xxi,j=x.Definition 8.
LetG be a connected graph. Then, the expanded χ−− chromatic Schultz polynomial of G, denoted by Sχ−G,x,k, is defined as
(13)Sχ−G,x,k=12∑i=1k∑u∈Vi−∑v∈Vi−ξφ−u+ξφ−vxidu,v+∑i=1k∑j=1:j≠ik∑u∈Vi−∑v∈Vj−ξφ−u+ξφ−vxi,jdu,vSχ−G,x=Sχ−G,x,kxi=xxi,j=x.Definition 9.
LetG be a connected graph. Then, the expanded χ−− chromatic Gutman polynomial of G, denoted by Gutχ−G,x,k, is defined as
(14)Gutχ−G,x,k=12∑i=1k∑u∈Vi−∑v∈Vi−ξφ−u×ξφ−vxidu,v+∑i=1k∑j=1:j≠ik∑u∈Vi−∑v∈Vj−ξφ−u×ξφ−vxi,jdu,vGutχ−G,x=Gutχ−G,x,kxi=xxi,j=x.The Jahangir graph class of graphs is named after the figure which appears on the tomb of Noureddine Muhammad Salim, known by his imperial name, Jahangir corresponding toJ2,8. Jahangir was the fourth Mughal Emperor who ruled from 1605 to 1627. His tomb is located 5 kilometer northwest of Lahore, Pakistan along the banks of the River Ravi. The Jahangir graph Jn,m [23] is a graph on nm+1 vertices and mn+1 edges for n≥2 and m≥3. Jahangir graph Jn,m consists of cycle Cnm with one additional vertex which is adjacent to m vertices of Cnm at the distance to each other.In [22], we see the chromatic Gutman polynomial of some cycle related graphs. In [24–29], we see the Wiener index and the Hosoya polynomial of Jn,m for n=2,3,4,5,6,7. In [30–32], we see Schultz’ polynomial, Gutman’s polynomial, Schultz’ index, and Gutman’s index of Jn,m for n=2,3,4.


## 2. Discussion and Results

In this paper we compute the chromatic Schultz and Gutman polynomials of Jahangir graphJn,m for n=2,3. Also, we suggest three new definitions; the first definition is the expanded Hosoya polynomial. The second definition is expanded chromatic Schultz polynomial. The third definition is expanded chromatic Gutman polynomial. We compute these definitions of Jahangir graph Jn,m for n=2,3.Theorem 10.
(15)Sχ+J2,m,x=2m2−6mx4+3m2−6mx3+m2+7mx2+9mx1+6m+4,Gutχ+J2,m,x=2m2−6mx4+2m2−4mx3+12m2+152mx2+6mx1+5m+4,H+J2,m,x,2=12mm−3x24+mm−2x1,23+12mm−1x12+2mx22+3mx1,21,Sχ+J2,m,x,2=2m−3x24+3mm−2x1,23+mm−1x12+8mx22++9mx1,21+6m+4,Gutχ+J2,m,x,2=2mm−3x24+2mm−2x1,23+12mm−1x12+8mx22+6mx1,21+5m+4.Proof.
LetV=v1,v2,v3,⋯,v2m,v2m+1 be the vertices of J2,m. It is obvious that χJ2,m=2 because C2m contains an even number of vertices, therefore χC2m=2 and vertex v2m+1 taking the same color of v2ifor1≤i≤m. Let c1,c2 be the two colors we use for coloring J2,m.
With respect toφ+, the vertices v1,v3,v5,⋯,v2m−1, get the color c1 and the vertices v2,v4,v6,⋯.v2m,v2m+1, get the color c2 because the number of vertices v2,v4,v6,⋯,v2m+1 is bigger than the number of vertices v1,v3,v5,⋯,v2m−1. We have V1+=v1,v3,v5,⋯,v2m−1,V2+=v2,v4,v6,⋯,v2m,v2m+1. As we can see in Figure 2, the black vertices take the color which is c1 and the white vertices take the color which is c2.Figure 2
Jahangir graphJ2,6.In Table1, we will show in the first column the possible distance between different pairs of vertices J2,m, in the second column, we will show the number of corresponding color pairs, and in the third column, we will show the number color pairs.Table 1
Shows the distance sequence in Jahangir graphJ2,m.
du,vColorNumber of pairs0c1,c1mc2,c2m+11c1,c23 m2c1,c11/2mm−1c2,c22 m3c1,c2mm−24c2,c21/2mm−3From the Table1, we have
(16)Sχ+J2,m,x=412mm−3x4+3mm−2x3+212mm−1+42mx2+33mx1+2m+4m+1=2m2−6mx4+3m2−6mx3+m2+7mx2+9mx1+6m+4,Gutχ+J2,m,x=412mm−3x4+2mm−2x3+12mm−1+42mx2+23mx1+m+4m+1=2m2−6mx4+2m2−4mx3+12m2+152mx2+6mx1+5m+4,H+J2.m,x,2=12mm−3x24+mm−2x1,23+12mm−1x12+2mx22+3mx1,21,Sχ+J2,m,x,2=2mm−3x24+3mm−2x1,23+mm−1x12+8mx22+9mx1,21+6m+4,Gutχ+J2.m,x,2=2mm−3x24+2mm−2x1,23+12mm−1x12+8mx22+6mx1,21+5m+4.Theorem 11.
(17)Sχ−J2,m,x=m2−3mx4+3m2−6mx3+2m2+2mx2+9mx1+6m+2,Gutχ−J2,m,x=12m2−32mx4+2m2−4mx3+2m2x2+6mx1+5m+1,H−J2,m,x,2=12mm−3x14+mm−2x1,23+12mm−2x22+2mx12+3mx1,21,Sχ−J2,m,x,2=mm−3x14+3mm−2x1,23+2mm−1x22+4mx12+9mx1,21+6m+2,Gutχ−J2,m,x,2=12mm−3x14+2mm−2x1,23+2mm−1x22+2mx12+6mx1,21+5m+1.Proof.
With respect toφ−, the vertices v1,v3,v5,⋯,v2m−1 get the color c2, and the vertices v2,v4,v6,⋯,v2m,v2m+1 get the color c1. We have V1−=v2,v4,v6,⋯,v2m,v2m+1,V2−=v1,v3,v5,⋯.,v2m−1. From the Table 1, we exchange each c1 with c2 and each c2 with c1, so we have
(18)Sχ−J2,m,x=212mm−3x4+3mm−2x3+412mm−1+22mx2+33mx1+4m+2m+1=m2−3mx4+3m2−6mx3+2m2+2mx2+9mx1+6m+2,Gutχ−J2,m,x=112mm−3x4+2mm−2x3+412mm−1+12mx2+23mx1+4m+1m+1=12m2−32mx4+2m2−4mx3+2m2x2+6mx1+5m+1,H−J2,m,x,2=12mm−3x14+mm−2x1,23+12mm−1x22+2mx12+3mx1,21,Sχ−J2,m,x,2=mm−3x14+3mm−2x1,23+2mm−1x22+4mx12+9mx1,21+6m+2,Gutχ−J2,m,x,2=12mm−3x14+2mm−2x1,23+2mm−1x22+2mx12+6mx1,21+5m+1.Theorem 12.
Form is even, we have
(19)Sχ+J3,m,x=10m2−25mx4+10m2−10mx3+5m22+39m2x2+37m2x1+15m+2,Gutχ+J3,m,x=25m22−31mx4+25m22−13mx3+25m28+85m4x2+41m2x1+39m2+1,H+J3,m,x,3=m2m−2x24+mm−3x2,34+m2m−2x34+m2m−2x23+m2x2,33+m2m−2x33+mx1,22+mx1,32+m28+5m4x22+m24x2,32+m28+5m4x32+m2x1,21+m2x1,31+3mx2,31,Sχ+J3,m,x,3=2mm−2x24+5mm−3x2,34+3mm−2x34+2mm−2x23+5m2x2,33+3mm−2x33+3mx1,22+4mx1,32+m22+5mx22+5m24x2,32+3m24+5m2x32+3m2x1,21+2mx1,31+15mx2,31+15m+2,Gutχ+J3,m,x,3=2mm−2x24+6mm−3x2,34+9m2m−2x34+2mm−2x23+6m2x2,33+9m2m−2x33+2mx1,22+3mx1,32+m22+5mx22+3m22x2,32+9m28+45m4x32+mx1,21+3m2x1,31+18mx2,31+39m2+1,Sχ−J3,m,x=6m2−15mx4+6m2−6mx3+3m22+33m2x2+27m2x1+9m+6,Gutχ−J3,m,x=9m22−11mx4+9m22−5mx3+9m28+61m4x2+21m2x1+15m2+9,H−J3,m,x,3=m2m−2x14+mm−3x1,24+m2m−2x24+m2m−2x13+m2x1,23+m2m−2x23+m28+5m4x12+m24x1,22+mx1,32+m28+5m4x22+mx2,32+3mx1,21+m2x1,31+m2x2,31,Sχ−J3,m,x,3=mm−2x14+3mm−3x1,24+2mm−2x24+mm−2x13+3m2x1,23+2mm−2x23+m24+5m2x12+3m24x1,22+4mx1,32+m22+5mx22+5mx2,32+9mx1,21+2mx1,31+5m2x2,31+9m+6,Gutχ−J3,m,x,3=m2m−2x14+3mm−3x1,24+2mm−2x24+m2m−2x13+2m2x1,23+2mm−2x23+m28+5m4x12+m22x1,22+3mx1,32+m22+5mx22+6mx2,32+6mx1,21+3m2x1,31+3mx2,31+15m2+9.Proof.
Let numberV=v1,v2,v3,⋯,v3m,v3m+1 be the vertices of J3,m. It is obvious that χJ3,m=3 because C3m contains an even number of vertices, therefore χC2m=2 and vertex v3m+1 taking a different color of color C3m. Let c1,c2,c3 be the three colors we use for coloring J3,m.
With respect toφ+, the vertices v3m+1 get the color c1, the vertices v2,v4,v6,⋯.v3m get the color c2, and the vertices v1,v3,v5,⋯.v3m−1 get the color c3. We have V1+=v3m+1,V2+=v2,v4,v6,⋯,v3m,,V3+=v1,v3,v5,⋯,v3m−1. As we can see in Figure 3, the black vertices take the color which is c3, the white vertices take the color which is c2, and the black and white vertices take the color c1.Figure 3
Jahangir graphJ3,4.From the Table2, we have
(20)Sχ+J3,m,x=4m2m−2+5mm−3+6m2m−2x4+4m2m−2+5m2+6m2m−2x3+3m+4m+4m28+5m4+5m24+6m28+5m4x2+3m2+4m2+53mx1+21+43m2+63m2=10m2−25mx4+10m2−10mx3+5m22+39m2x2+37m2x1+15m+2,Gutχ+J3,m,x=4m2m−2+6mm−3+9m2m−2x4+4m2m−2+6m2+9m2m−2x3+2m+3m+4m28+5m4+6m24+9m28+5m4x2+2m2+3m2+63mx1+11+43m2+93m2=25m22−31mx4+25m22−13mx3+25m28+85m4x2+41m2x1+39m2+1,H+J3,m,x,3=m2m−2x24+mm−3x2,34+m2m−2x34+m2m−2x23+m2x2,33+m2m−2x33+mx1,22+mx1,32+m28+5m4x22+m24x2,32+m28+5m4x32+m2x1,21+m2x1,31+3mx2,31,Sχ+J3,m,x,3=2mm−2x24+5mm−3x2,34+3mm−2x34+2mm−2x23+5m2x2,33+3mm−2x33+3mx1,22+4mx1,32+m22+5mx22+5m24x2,32+3m24+5m2x32+3m2x1,21+2mx1,31+15mx2,31+15m+2,Gutχ+J3,m,x,3=2mm−2x24+6mm−3x2,34+9m2m−2x34+2mm−2x23+6m2x2,33+9m2m−2x33+2mx1,22+3mx1,32+m22+5mx22+3m22x2,32+9m28+45m4x32+mx1,21+3m2x1,31+18mx2,31+39m2+1.Table 2
Shows the distance sequence in Jahangir graphJ3,m for m is even.
du,vColorNumber of pairs0c1,c11c2,c23m/2c3,c33m/21c1,c2m/2c1,c3m/2c2,c33m2c1,c2mc1,c3mc2,c2m2/8+5m/4c2,c3m2/4c3,c3m2/8+5m/43c2,c2m/2m−2c2,c3m2c3,c3m/2m−24c2,c2m/2m−2c2,c3mm−3c3,c3m/2m−2With respect toφ−, the vertices v3m+1 get the color c3, the vertices v2,v4,v6,⋯.v3m get the color c2, and the vertices v1,v3,v5,⋯.v3m−1 get the color c1. We have V1−=v1,v3,v5,⋯,v3m−1,V2−=v2,v4,v6,⋯,v3m,V3−=v3m+1. From the Table 2, we exchange each c1 with c3 and each c3 with c1, so we have
(21)Sχ−J3,m,x=2m2m−2+3mm−3+4m2m−2x4+2m2m−2+3m2+4m2m−2x3+2m28+5m4+3m24+4m+4m28+5m4+5mx2+33m+4m2+5m2x1+23m2+43m2+61=6m2−15mx4+6m2−6mx3+3m22+33m2x2+27m2x1+9m+6,Gutχ−J3,m,x=1m2m−2+2mm−3+4m2m−2x4+1m2m−2+4m2m−2+2m2x3+1m28+5m4+2m24+3m+6m+4m28+5m4x2+23m+3m2+6m2x1+13m2+43m2+91=9m22−11mx4+9m22−5mx3+9m28+61m4x2+21m2x1+15m2+9,H−J3,m,x,3=m2m−2x14+mm−3x1,24+m2m−2x24+m2m−2x13+m2x1,23+m2m−2x23+m28+5m4x12+m24x1,22+mx1,32+m28+5m4x22+mx2,32+3mx1,21+m2x1,31+m2x2,31,Sχ−J3,m,x,3=mm−2x14+3mm−3x1,24+2mm−2x24+mm−2x13+3m2x1,23+2mm−2x23+m24+5m2x12+3m24x1,22+4mx1,32+m22+5mx22+5mx2,32+9mx1,21+2mx1,31+5m2x2,31+9m+6,Gutχ−J3,m,x,3=m2m−2x14+3mm−3x1,24+2mm−2x24+m2m−2x13+2m2x1,23+2mm−2x23+m28+5m4x12+m22x1,22+3mx1,32+m22+5mx22+6mx2,32+6mx1,21+3m2x1,31+3mx2,31+15m2+9.Theorem 13.
Form is odd, we have
(22)Sχ+J3,m,x=10m2−29m+10x4+10m2−11m−2x3+5m22+20m−112x2+37m2−52x1+15m−1,Gutχ+J3,m,x=25m22−41m+472x4+25m22−31m2−3x3+25m28+45m2−938x2+41m2−132x1+39m2−92,H+J3,m,x,3=m−3x1,24+m−2x1,34+m2−2m−12x24+m2−4m+4x2,34+m2−4m+32x34+m+12x1,23+m−12x1,33+m2−3m+42x23+2m2−m−52x2,33+m2−2m+12x33+x12+m+1x1,22+mx1,32+m2+8m−98x22+m2+34x2,32+m2+12m−138x32+m+12x1,21+m+32x1,31+3m−2x2,31,Sχ+J3,m,x,3=3m−3x1,24+4m−2x1,34+2m2−2m−1x24+5m2−4m+4x2,34+3m2−4m+3x34+3m+32x1,23+2m−1x1,33+2m2−3m+4x23+10m2−5m−252x2,33+3m2−2m+1x33+2x12+3m+1x1,22+4mx1,32+m2+8m−92x22+5m2+154x2,32+3m2+36m−394x32+3m+32x1,21+2m+3x1,31+53m−2x2,31+15m−1,Gutχ+J3,m,x,3=2m−3x1,24+3m−2x1,34+2m2−2m−1x24+6m2−4m+4x2,34+9m2−36m+272x34+m+1x1,23+3m−32x1,33+2m2−3m+4x23+32m2−m−5x2,33+9m2−18m+92x33+x12+2m+1x1,22+3mx1,32+m2+8m−92x22+3m2+92x2,32+9m2+108m−1178x32+m+1x1,21+3m+92x1,31+63m−2x2,31+39m2−92,Sχ−J3,m,x=6m2−11m−10x4+6m2−5m+2x3+3m22+16m+12x2+27m2+52x1+9m+9.Gutχ−J3,m,x=9m22−5m−332x4+9m22−7m2+5x3+9m22+29m2+838x2+21m2+72x1+15m2+312,H−J3,m,x,3=m2−4m+32x14+m2−4m+4x1,24+m−2x1,34+m2−2m−12x24+m−3x2,34+m2−2m+12x13+2m2−m−52x1,23+m−12x1,33+m2−3m+42x23+m+12x2,33+m2+12m−138x12+m2+34x1,22+mx1,32+m2+8m−98x22+m+1x2,32+x3,32+3m−2x1,21+m+32x1,31+m+12x2,31,Sχ−J3,m,x,3=m2−4m+3x14+3m2−4m+4x1,24+4m−2x1,34+2m2−2m−1x24+5m−3x2,34+m2−2m+1x13+6m2−3m−152x1,23+2m−1x1,33+2m2−3m+4x23+5m+52x2,33+m2+12m−134x12+3m2+94x1,22+4mx1,32+m2+8m−92x22+5m+1x2,32+6x3,32+33m−2x1,21+2m+3x1,31+5m+52x2,31+9m+9,Gutχ−J3,m,x,3=m2−4m+32x14+2m2−4m+4x1,24+3m−2x1,34+2m2−2m−1x24+6m−3x2,34+m2−2m+12x13+2m2−m−5x1,23+3m−32x1,33+2m2−3m+4x23+3m+1x2,33+m2+12m−138x12+m2+32x1,22+3mx1,32+m2+8m−92x22+6m+1x2,32+9x3,32+23m−2x1,21+3m+92x1,31+3m+1x2,31+15m2+312.Proof.
Let numberV=v1,v2,v3,⋯,v3m,v3m+1 be the vertices of J3,m. It is obvious that χJ3,m=3 because C3m contains an odd number of vertices, therefore χC3m=3 and vertex v3m+1 are taking the same color of v3m. Let c1,c2,c3 be the three color we use for coloring J3,m.
With respect toφ+, the vertices v3m+1,v3m get the color c1, the vertices v2,v4,v6,⋯.v3m−1 get the color c2, and the vertices v1,v3,v5,⋯.v3m−2 get the color c3. We have V1+=v3m+1,v3m,V2+=v2,v4,⋯,v3m−1,V3+=v1,v3,⋯,v3m−2. As we can see in Figure 4, the black vertices take the color which is c3, the white vertices take the color which is c2, and the black and white vertices take the color c1.Figure 4
Jahangir graphJ3,5.From the Table3, we have
(23)Sχ+J3,m,x=3m−3+4m−2+4m2−2m−12+5m2−4m+4+6m2−4m+32x4+3m+12+4m−12+4m2−3m+42+52m2−m−52+6m2−2m+12x3+21+3m+1+4m+4m2+8m−98+5m2+34+6m2+12m−138x2+3m+12+4m+32+53m−2x1+22+43m−12+63m−12=10m2−29m+10x4+10m2−11m−2x3+5m22+20m−112x2+37m2−52x1+15m−1,Gutχ+J3,m,x=2m−3+3m−2+4m2−2m−12+6m2−4m+4+9m2−4m+32x4+2m+12+3m−12+4m2−3m+42+62m2−m−52+9m2−2m+12x3+11+2m+1+3m+4m2+8m−98+6m2+34+9m2+12m−138x2+2m+12+3m+32+63m−2x1+12+43m−12+93m−12=25m22−41m+472x4+25m22−31m2−3x3+25m28+45m2−938x2+41m2−132x1+39m2−92,H+J3,m,x,3=m−3x1,24+m−2x1,34+m2−2m−12x24+m2−4m+4x2,34+m2−4m+32x34+m+12x1,23+m−12x1,33+m2−3m+42x23+2m2−m−52x2,33+m2−2m+12x33+x12+m+1x1,22+mx1,32+m2+8m−98x22+m2+34x2,32+m2+12m−138x32+m+12x1,21+m+32x1,31+3m−2x2,31,Sχ+J3,m,x,3=3m−3x1,24+4m−2x1,34+2m2−2m−1x24+5m2−4m+4x2,34+3m2−4m+3x34+3m+32x1,23+2m−1x1,33+2m2−3m+4x23+10m2−5m−252x2,33+3m2−2m+1x33+2x12+3m+1x1,22+4mx1,32+m2+8m−92x22+5m2+154x2,32+3m2+36m−394x32+3m+32x1,21+2m+3x1,31+53m−2x2,31+15m−1,Gutχ+J3,m,x,3=2m−3x1,24+3m−2x1,34+2m2−2m−1x24+6m2−4m+4x2,34+9m2−36m+272x34+m+1x1,23+3m−32x1,33+2m2−3m+4x23+32m2−m−5x2,33+9m2−18m+92x33+x12+2m+1x1,22+3mx1,32+m2+8m−92x22+3m2+92x2,32+9m2+108m−1178x32+m+1x1,21+3m+92x1,31+63m−2x2,31+39m2−92.Table 3
Shows the distance sequence in Jahangir graphJ3,m for m is odd.
du,vColorNumber of pairs0c1,c12c2,c23m−1/2c3,c33m−1/21c1,c2m+1/2c1,c3m+3/2c2,c33m−22c1,c11c1,c2m+1c1,c3mc2,c2m2+8m−9/8c2,c3m2+3/4c3,c3m2+12m−13/83c1,c2m+1/2c1,c3m−1/2c2,c2m2−3m+4/2c2,c32m2−m−5/2c3,c3m2−2m+1/24c1,c2m−3c1,c3m−2c2,c2m2−2m−1/2c2,c3m2−4m+4c3,c3m2−4m+3/2With respect toφ−, the vertices v3m+1,v3m get the color c3, the vertices v2,v4,v6,⋯.v3m−1 get the color c2, and the vertices v1,v3,v5,⋯.v3m−2 get the color c1. We have V1−=v1,v3,⋯,v3m−2,V2−=v2,v4,⋯,v3m−1,V3−=v3m,v3m+1. From the Table 3, we exchange each c1 with c3 and each c3 with c1, so we have
(24)Sχ−J3,m,x=2m2−4m+32+3m2−4m+4+4m2−2m−12+4m−2+5m−3x4+2m2−2m+12+32m2−m−52+4m−12+4m2−3m+42+5m+12x3+2m2+12m−138+3m2+34+4m+4m2+8m−98+5m+1+11x2+33m−2+4m+32+5m+12x1+23m−12+43m−12+62=6m2−11m−10x4+6m2−5m+2x3+3m22+16m+12x2+27m2+52x1+9m+9,Gutχ−J3,m,x=1m2−4m+32+2m2−4m+4+3m−2+4m2−2m−12+6m−3x4+1m2−2m+12+22m2−m−52+3m−12+4m2−3m+42+6m+12x3+1m2+12m−138+2m2+34+3m+4m2+8m−98+6m+1+91x2+23m−2+3m+32+6m+12x1+13m−12+43m−12+92=9m22−5m−332x4+9m22−7m2+5x3+9m22+29m2+838x2+21m2+72x1+15m2+312,H−J3,m,x,3=m2−4m+32x14+m2−4m+4x1,24+m−2x1,34+m2−2m−12x24+m−3x2,34+m2−2m+12x13+2m2−m−52x1,23+m−12x1,33+m2−3m+42x23+m+12x2,33+m2+12m−138x12+m2+34x1,22+mx1,32+m2+8m−98x22+m+1x2,32+x3,32+3m−2x1,21+m+32x1,31+m+12x2,31,Sχ−J3,m,x,3=m2−4m+3x14+3m2−4m+4x1,24+4m−2x1,34+2m2−2m−1x24+5m−3x2,34+m2−2m+1x13+6m2−3m−152x1,23+2m−1x1,33+2m2−3m+4x23+5m+52x2,33+m2+12m−134x12+3m2+94x1,22+4mx1,32+m2+8m−92x22+5m+1x2,32+6x3,32+33m−2x1,21+2m+3x1,31+5m+52x2,31+9m+9,Gutχ−J3,m,x,3=m2−4m+32x14+2m2−4m+4x1,24+3m−2x1,34+2m2−2m−1x24+6m−3x2,34+m2−2m+12x13+2m2−m−5x1,23+3m−32x1,33+2m2−3m+4x23+3m+1x2,33+m2+12m−138x12+m2+32x1,22+3mx1,32+m2+8m−92x22+6m+1x2,32+9x3,32+23m−2x1,21+3m+92x1,31+3m+1x2,31+15m2+312.


## 3. Conclusion

In this paper, we computed the chromatic Schultz and Gutman polynomials of Jahangir graphJn,m for n=2,3. Also, we suggested three new definitions; the first definition is the expanded Hosoya polynomial, where there are expanded H−− Hosoya polynomial and expanded H+− Hosoya polynomial. The second definition is the expanded chromatic Schultz polynomial, where there are expanded χ+− chromatic Schultz polynomial and expanded χ−− chromatic Schultz polynomial. The third definition is the expanded chromatic Gutman polynomial where there are expanded χ+− chromatic Gutman polynomial and expanded χ−− chromatic Gutman polynomial. We computed these suggested definitions of Jahangir graph Jn,m for n=2,3. The importance of these suggested definitions is that they give the colors of vertices in the beginning and the end of every path in the graph.



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*Source: 4891083-2023-01-04.xml*