# Minimum Detour Index of Tricyclic Graphs

Authors: Wei Fang; Zheng-Qun Cai; Xiao-Xin Li
Journal: Journal of Chemistry
 (2019)

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## Abstract

The detour index of a connected graph is defined as the sum of the detour distances (lengths of longest paths) between unordered pairs of vertices of the graph. The detour index is used in various quantitative structure-property relationship and quantitative structure-activity relationship studies. In this paper, we characterize the minimum detour index among all tricyclic graphs, which attain the bounds.


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## Body

## 1. Introduction

LetG be a simple and connected graph with VG=n and EG=m and NGu be the neighbor vertex set of vertex u, then dGu=NGu is called the degree of u. If m=n−1+c, then G is called a c-cyclic graph. If c=0,1,2, and 3, then G is a tree, unicyclic graph, bicyclic graph, and tricyclic graph, respectively. Denote by Tn the set of all tricyclic graphs of order n.LetT^=T^i∣1≤i≤15, where graphs T^i for i=1,2,…,15 are defined in Figure 1. By [1, 2], we know that for any Ti∈Tn, Ti can be obtained from T^i1≤i≤15 by attaching trees to some of its vertices. We call T^i the base of Ti.Figure 1
The fifteen types of bases for tricyclic graphs.A block of the graphG is a maximal 2-connected subgraph of G. A cactus is a connected graph in which no edge lies in more than one cycle, such that each block of a cactus is either an edge or a cycle. A vertex shared by two or more cycles is called a cut vertex. In this paper, denote Cnl be the set of all cacti of order n and l cycles, where l≥1. The length of the cycles may be different and the length of each cycle is at least 3.The concept of “topological index” was first proposed by Haruo Hosoya for characterizing the topological nature of a graph. Such graph invariants are usually related to the distance functiond−,−.The detour distance [3, 4] (also known under the name elongation) between vertices u and v in G is the length of a longest path between them, denoted by lu,v∣G. Note that lu,u∣G=0 for any u∈VG; see [5] for a discussion. The detour index of the graph G is defined as [4–9](1)ωG=12∑u,v∈VGlu,v∣G.For a connected graphG with u∈VG, let Lu∣G=∑v∈VGlu,v∣G, then(2)ωG=12∑u∈VGLu∣G.If we use the notion of the detour matrix [4], which is an n×n matrix whose i,j-element is lvi,vj∣G with VG=v1,v2,…,vn, then the detour index is equal to the half-sum of the (off diagonal) elements of the detour matrix. The detour index has been applied to chemistry, especially in quantitative structure-activity relationship (QSAR) studies; see [7, 10] for more details. A new branch cheminformatics is a combination of mathematics and chemistry. This branch studies QSAR/QSPR study, physicochemical properties and topological indices such as Zagreb Indices [11], Kirchhoff index [12], Hosoya index [13] and so on to predict physicochemical properties and biological activities of the chemical compounds theoretically.In this paper, we consider the minimum detour index among all tricyclic graphs.


## 2. Preliminaries

In this section, we will introduce some useful lemmas and graph transformations.


### 2.1. Edge-Lifting Transformation [14, 15]

LetG1 and G2 be two graphs with n1≥2 and n2≥2 vertices, respectively. If G is the graph obtained from G1 and G2 by adding an edge between a vertex u0 of G1 and a vertex v0 of G2, G′ is the graph obtained by identifying u0 of G1 to v0 of G2 and adding a pendant edge to u0v0, then G′ is called the edge-lifting transformation of G (see Figure 2).Figure 2
The edge-lifting transformation.Lemma 1 (see [16]).
Let G be defined as in Figure2, and G′ is obtained from G by the edge-lifting transformation (see Figure 2). Then, ωG>ωG′.DenoteTn1=T1,T7,T8,T9,T12,T13,T14,T15 (see Figure 1).By Lemma 1, we can verify that ifT∈Tn attains the minimum detour index of all graphs in Tn, then the following two conditions hold:(i)
The baseT^ of T is one of T^n1(ii)
The graphT is obtained from T^ by attaching some pendant edgesRemark 1.
In order to determine the tricyclic graphs which attain the minimum detour index of all graphs inTn, we just need to discuss the tricyclic graphs in T, where T^∈Tn1.


### 2.2. Cycle-Edge Transformation

LetC∈Cnl be a cactus as shown in Figure 3, where Cr=v1v2…vrv1 is the biggest cycle of C, r≥4. Denote the vertex set Wvi=NGivi=NGvi∩VGi,1≤i≤r. C′ is the graph obtained from C by deleting the edges v2v3 and v2 to Wv2, meanwhile adding the edges v1v3 and v1 to Wv2.Figure 3
The cycle-edge transformation.We say thatC′ is obtained from C by the cycle-edge transformation (see Figure 3).Lemma 2 (see [16]).
LetC∈Cnl be a cactus as shown in Figure 3 with r≥4, and C′ be the cycle-edge transformation of C (see Figure 3). Then, ωC>ωC′.


### 2.3. Cycle-Lifting Transformation

LetC1∈Cnl be a cactus as shown in Figure 4. Denote Wvi=NGivi=NGvi∩VGi for 1≤i≤3. Let C1′ be the graph obtained from C1 by deleting the edges v2vx for vx∈Wv2 and adding the edges v1vx for vx∈Wv2.Figure 4
The cycle-lifting transformation.We say thatC1′ is the cycle-lifting transformation of C1 (see Figure 4).Lemma 3 (see [16]).
LetC1′ be the cycle-lifting transformation of C1 (see Figure 4). Then, ωC1>ωC1′.


### 2.4. Operation I

We define Operation I as follows. LetG and G′ be a simple and connected graph as shown in Figure 5. v1v2…vp be the path in a cycle. Denote Wvi=w∣wvi∈EGanddw=1,1≤i≤p,p≥3, and G′ be the graph obtained from G by deleting the edges v2v3, v2w for w∈Wv2 and adding the edges v1v3,v1w for w∈Wv2 (see Figure 5).Figure 5
Operation I on graphG.Lemma 4.
LetG and G′ be the graph shown in Figure 5. Then, ωG>ωG′.Proof.
LetVG=VG′=v1,v2,v3,…,vn, and Wvi=w∣wvi∈EGanddw=1,1≤i≤p. For the vertices vi,vj∈VG−v2, obviously(3)lvi,vjG≥lvi,vjG′,(4)lv1,v2G>lv1,v2G′=1.
LetPi be the set of the longest path between v2 and vi in G and Qi be the set of the longest path between v1 and vi in G, where 3≤i≤n.
Case 1.
v
1
v
2
∈
E
P
i and v2v3∈EPi, where 3≤i≤n.
Obviously, for3≤i≤n, lv2,viG=lv2,viG′−1. On the other hand, if v1v2∈EPi and v2v3∈EPi, then v1v2,v2v3∈ℒi, where ℒi be the any one longest path between v1 and vi in G. Therefore, lv1,viG=lv1,viG′+1, and(5)lv2,viG+lv1,viG=lv2,viG′+lv1,viG′,3≤i≤n.
Case 2.
v
1
v
2
∉
E
P
i and v2v3∈EPi.
Obviously, for3≤i≤n, we have(6)lv2,vi∣G=lv3,viG+1>lv1,vi∣G+1≥lv1,vi∣G′+1=lv2,vi∣G′.
Case 3.
v
1
v
2
∈
E
P
i and v2v3∉EPi.
Obviously, for3≤i≤n, we have(7)lv2,viG=lv2,viG′.
Case 4.
v
1
v
2
∉
E
P
i and v2v3∉EPi.
Obviously,vi∈Wv2⊂VG, and(8)lv2,viG=lv1,viG′=1,(9)lv1,viG>lv2,viG′=2.
By (5)–(9), we have(10)lv2,viG+lv1,viG≥lv2,viG′+lv1,viG′,3≤i≤n.
By (3), (4), and (10), we have ωG>ωG′.


### 2.5. Operation II

We define Operation II as follows. LetG and G′ be a simple and connected graph as shown in Figure 6. Denote v1v2v3v1 be a cycle with length 3, Wvi=w∣wvi∈EGanddw=1,i=1,2, and G′ be the graph obtained from G by deleting the edges v2w for w∈Wv2 and adding the edges v1w for w∈Wv2 (see Figure 6).Figure 6
Operation II on graphG.Lemma 5.
LetG and G′ be the graph shown in Figure 6. Then, ωG>ωG′, and the equality holds if and only if G≅G′.Proof.
LetVG=VG′=v1,v2,v3,…,vn. For the vertices vi,vj∈VG−Wv2, vx,vy∈Wv2, we have(11)lvi,vj∣G=lvi,vjG′,(12)lvx,vy∣G=lvx,vyG′=2.
For the verticesvi∈VG−Wv2−v1,v2, w∈Wv2, we have(13)lw,viG=max3+lv1,viG,3+lv3,viG≥max1+lv1,viG,3+lv3,viG=max1+lv1,viG′,3+lv3,viG′=lw,vi∣G′,especially, if G≇G′, then(14)lw,v3G>lw,v3G′,(15)lw,v1G=lv1,v2G+1=lv1,v2∣G′+1=lw,v2∣G′,(16)lw,v2G=lw,v1G′=1.
By (11)–(16), we have ωG≥ωG′, and the equality holds if and only if G≅G′.DenoteTn2=T16,T17,T18,T19,T20,T21; see Figure 7.By Lemma 2–5, we can verify that ifT∈Tn attains the minimum detour index of all graphs in Tn, then T∈Tn2.Figure 7
GraphTn2.Remark 2.
In order to determine the tricyclic graphs which attain the minimum detour index of all graphs inTn, we just need to discuss the tricyclic graphs in T, where T^∈Tn2=T16,T17,T18,T19,T20,T21; see Figure 7.


### 2.6. Operation III

We define Operation III as follows. LetG∈T17∪T18∪T20∪T21 as shown in Figure 7. Denote Wvi=w∣wvi∈EGanddw=1 and G′ be the graph obtained from G by deleting the edges v3w for w∈Wv3 and adding the edges v1w for w∈Wv3 (see Figures 8–11).Figure 8
Operation III on graphG1∈T17.Figure 9
Operation III on graphG2∈T18.Figure 10
Operation III on graphG3∈T20.Figure 11
Operation III on graphG4∈T21.Lemma 6.
LetGi and Gi′1≤i≤4 be the graph in Figures 8–11. Then, ωG≥ωG′, and the equality holds if and only if G≅G′.Proof.
LetVG1=VG1′=v1,v2,v3,…,vn. For the vertices vi,vj∈VG−Wv3−v1,v3; vx,vy∈Wv3, we have(17)lvi,vjG1=lvi,vjG1′,(18)lvx,vyG1=lvx,vyG1′=2,(19)lv1,viG1=lv1,viG1′,(20)lv3,vxG1=lv1,vxG1′,(21)lv3,viG1≥lv1,viG1′,(22)lvx,viG1=lv3,viG1+1=lv3,vi∣G1′+1≥lv1,vi∣G1′+1=lvx,vi∣G1′,(23)lvx,v5G1=5>3=lvx,v5G1′.
By (17)–(23), we have ωG1>ωG1′.
Similarly, we haveωGi≥ωGi′, and the equality holds if and only if Gi≅Gi′ (i = 2, 3, 4).


### 2.7. Operation IV

We define Operation IV as follows. LetG∈T19 as shown in Figure 12. Denote Wvi=w∣wvi∈EGanddw=1, and G′ be the graph obtained from G by deleting the edges viw for w∈Wvi,i=2,4,5 and adding the edges v1w for w∈Wvi,i=2,4,5 (see Figure 12).Figure 12
Operation IV on graphG∈T19.Lemma 7.
LetG and G′ be the graph shown in Figure 12. Then, ωG≥ωG′ with equality holding if and only if G≅G′.Proof.
LetV1=vi∣1≤i≤6, V2=VG−V1, we have(24)ωG=∑vx,vy∈V1lvx,vy∣G+∑vx∈V1,vy∈V2lvx,vy∣G+∑vx,vy∈V2lvx,vy∣G,(25)ωG′=∑vx,vy∈V1lvx,vy∣G′+∑vx∈V1,vy∈V2lvx,vy∣G′+∑vx,vy∈V2lvx,vy∣G′.
Obviously,(26)∑vx,vy∈V1lvx,vy∣G=∑vx,vy∈V1lvx,vy∣G′,∑vx∈V1,vy∈V2lvx,vy∣G=+∑vx∈V1,vy∈V2lvx,vy∣G′,(27)lvx,vyG=lvx,vyG′=2,vx,vy∈Wvi,lvx,vyG>2=lvx,vyG′,vx∈Wvi,vy∈Wvj,i≠j.
Therefore,(28)ωG−ωG′=∑vx,vy∈V2lvx,vy∣G−∑vx,vy∈V2lvx,vy∣G′≥0,and the equality holds if and only if G≅G′.DenoteTn3=T16∪T22∪T23∪T24∪T25∪T26 (see Figure 13).By Lemma 6-7, we can verify that ifT∈Tn attains the minimum detour index of all graphs in Tn, then T^∈Tn3.Figure 13
GraphTn3.Remark 3.
In order to determine the tricyclic graphs which attain the minimum detour index of all graphs inTn, we just need to discuss the tricyclic graphs in T, where T^∈Tn3=T16,T22,T23,T24,T25,T26 (see Figure 13).


## 2.1. Edge-Lifting Transformation [14, 15]

LetG1 and G2 be two graphs with n1≥2 and n2≥2 vertices, respectively. If G is the graph obtained from G1 and G2 by adding an edge between a vertex u0 of G1 and a vertex v0 of G2, G′ is the graph obtained by identifying u0 of G1 to v0 of G2 and adding a pendant edge to u0v0, then G′ is called the edge-lifting transformation of G (see Figure 2).Figure 2
The edge-lifting transformation.Lemma 1 (see [16]).
Let G be defined as in Figure2, and G′ is obtained from G by the edge-lifting transformation (see Figure 2). Then, ωG>ωG′.DenoteTn1=T1,T7,T8,T9,T12,T13,T14,T15 (see Figure 1).By Lemma 1, we can verify that ifT∈Tn attains the minimum detour index of all graphs in Tn, then the following two conditions hold:(i)
The baseT^ of T is one of T^n1(ii)
The graphT is obtained from T^ by attaching some pendant edgesRemark 1.
In order to determine the tricyclic graphs which attain the minimum detour index of all graphs inTn, we just need to discuss the tricyclic graphs in T, where T^∈Tn1.


## 2.2. Cycle-Edge Transformation

LetC∈Cnl be a cactus as shown in Figure 3, where Cr=v1v2…vrv1 is the biggest cycle of C, r≥4. Denote the vertex set Wvi=NGivi=NGvi∩VGi,1≤i≤r. C′ is the graph obtained from C by deleting the edges v2v3 and v2 to Wv2, meanwhile adding the edges v1v3 and v1 to Wv2.Figure 3
The cycle-edge transformation.We say thatC′ is obtained from C by the cycle-edge transformation (see Figure 3).Lemma 2 (see [16]).
LetC∈Cnl be a cactus as shown in Figure 3 with r≥4, and C′ be the cycle-edge transformation of C (see Figure 3). Then, ωC>ωC′.


## 2.3. Cycle-Lifting Transformation

LetC1∈Cnl be a cactus as shown in Figure 4. Denote Wvi=NGivi=NGvi∩VGi for 1≤i≤3. Let C1′ be the graph obtained from C1 by deleting the edges v2vx for vx∈Wv2 and adding the edges v1vx for vx∈Wv2.Figure 4
The cycle-lifting transformation.We say thatC1′ is the cycle-lifting transformation of C1 (see Figure 4).Lemma 3 (see [16]).
LetC1′ be the cycle-lifting transformation of C1 (see Figure 4). Then, ωC1>ωC1′.


## 2.4. Operation I

We define Operation I as follows. LetG and G′ be a simple and connected graph as shown in Figure 5. v1v2…vp be the path in a cycle. Denote Wvi=w∣wvi∈EGanddw=1,1≤i≤p,p≥3, and G′ be the graph obtained from G by deleting the edges v2v3, v2w for w∈Wv2 and adding the edges v1v3,v1w for w∈Wv2 (see Figure 5).Figure 5
Operation I on graphG.Lemma 4.
LetG and G′ be the graph shown in Figure 5. Then, ωG>ωG′.Proof.
LetVG=VG′=v1,v2,v3,…,vn, and Wvi=w∣wvi∈EGanddw=1,1≤i≤p. For the vertices vi,vj∈VG−v2, obviously(3)lvi,vjG≥lvi,vjG′,(4)lv1,v2G>lv1,v2G′=1.
LetPi be the set of the longest path between v2 and vi in G and Qi be the set of the longest path between v1 and vi in G, where 3≤i≤n.
Case 1.
v
1
v
2
∈
E
P
i and v2v3∈EPi, where 3≤i≤n.
Obviously, for3≤i≤n, lv2,viG=lv2,viG′−1. On the other hand, if v1v2∈EPi and v2v3∈EPi, then v1v2,v2v3∈ℒi, where ℒi be the any one longest path between v1 and vi in G. Therefore, lv1,viG=lv1,viG′+1, and(5)lv2,viG+lv1,viG=lv2,viG′+lv1,viG′,3≤i≤n.
Case 2.
v
1
v
2
∉
E
P
i and v2v3∈EPi.
Obviously, for3≤i≤n, we have(6)lv2,vi∣G=lv3,viG+1>lv1,vi∣G+1≥lv1,vi∣G′+1=lv2,vi∣G′.
Case 3.
v
1
v
2
∈
E
P
i and v2v3∉EPi.
Obviously, for3≤i≤n, we have(7)lv2,viG=lv2,viG′.
Case 4.
v
1
v
2
∉
E
P
i and v2v3∉EPi.
Obviously,vi∈Wv2⊂VG, and(8)lv2,viG=lv1,viG′=1,(9)lv1,viG>lv2,viG′=2.
By (5)–(9), we have(10)lv2,viG+lv1,viG≥lv2,viG′+lv1,viG′,3≤i≤n.
By (3), (4), and (10), we have ωG>ωG′.


## 2.5. Operation II

We define Operation II as follows. LetG and G′ be a simple and connected graph as shown in Figure 6. Denote v1v2v3v1 be a cycle with length 3, Wvi=w∣wvi∈EGanddw=1,i=1,2, and G′ be the graph obtained from G by deleting the edges v2w for w∈Wv2 and adding the edges v1w for w∈Wv2 (see Figure 6).Figure 6
Operation II on graphG.Lemma 5.
LetG and G′ be the graph shown in Figure 6. Then, ωG>ωG′, and the equality holds if and only if G≅G′.Proof.
LetVG=VG′=v1,v2,v3,…,vn. For the vertices vi,vj∈VG−Wv2, vx,vy∈Wv2, we have(11)lvi,vj∣G=lvi,vjG′,(12)lvx,vy∣G=lvx,vyG′=2.
For the verticesvi∈VG−Wv2−v1,v2, w∈Wv2, we have(13)lw,viG=max3+lv1,viG,3+lv3,viG≥max1+lv1,viG,3+lv3,viG=max1+lv1,viG′,3+lv3,viG′=lw,vi∣G′,especially, if G≇G′, then(14)lw,v3G>lw,v3G′,(15)lw,v1G=lv1,v2G+1=lv1,v2∣G′+1=lw,v2∣G′,(16)lw,v2G=lw,v1G′=1.
By (11)–(16), we have ωG≥ωG′, and the equality holds if and only if G≅G′.DenoteTn2=T16,T17,T18,T19,T20,T21; see Figure 7.By Lemma 2–5, we can verify that ifT∈Tn attains the minimum detour index of all graphs in Tn, then T∈Tn2.Figure 7
GraphTn2.Remark 2.
In order to determine the tricyclic graphs which attain the minimum detour index of all graphs inTn, we just need to discuss the tricyclic graphs in T, where T^∈Tn2=T16,T17,T18,T19,T20,T21; see Figure 7.


## 2.6. Operation III

We define Operation III as follows. LetG∈T17∪T18∪T20∪T21 as shown in Figure 7. Denote Wvi=w∣wvi∈EGanddw=1 and G′ be the graph obtained from G by deleting the edges v3w for w∈Wv3 and adding the edges v1w for w∈Wv3 (see Figures 8–11).Figure 8
Operation III on graphG1∈T17.Figure 9
Operation III on graphG2∈T18.Figure 10
Operation III on graphG3∈T20.Figure 11
Operation III on graphG4∈T21.Lemma 6.
LetGi and Gi′1≤i≤4 be the graph in Figures 8–11. Then, ωG≥ωG′, and the equality holds if and only if G≅G′.Proof.
LetVG1=VG1′=v1,v2,v3,…,vn. For the vertices vi,vj∈VG−Wv3−v1,v3; vx,vy∈Wv3, we have(17)lvi,vjG1=lvi,vjG1′,(18)lvx,vyG1=lvx,vyG1′=2,(19)lv1,viG1=lv1,viG1′,(20)lv3,vxG1=lv1,vxG1′,(21)lv3,viG1≥lv1,viG1′,(22)lvx,viG1=lv3,viG1+1=lv3,vi∣G1′+1≥lv1,vi∣G1′+1=lvx,vi∣G1′,(23)lvx,v5G1=5>3=lvx,v5G1′.
By (17)–(23), we have ωG1>ωG1′.
Similarly, we haveωGi≥ωGi′, and the equality holds if and only if Gi≅Gi′ (i = 2, 3, 4).


## 2.7. Operation IV

We define Operation IV as follows. LetG∈T19 as shown in Figure 12. Denote Wvi=w∣wvi∈EGanddw=1, and G′ be the graph obtained from G by deleting the edges viw for w∈Wvi,i=2,4,5 and adding the edges v1w for w∈Wvi,i=2,4,5 (see Figure 12).Figure 12
Operation IV on graphG∈T19.Lemma 7.
LetG and G′ be the graph shown in Figure 12. Then, ωG≥ωG′ with equality holding if and only if G≅G′.Proof.
LetV1=vi∣1≤i≤6, V2=VG−V1, we have(24)ωG=∑vx,vy∈V1lvx,vy∣G+∑vx∈V1,vy∈V2lvx,vy∣G+∑vx,vy∈V2lvx,vy∣G,(25)ωG′=∑vx,vy∈V1lvx,vy∣G′+∑vx∈V1,vy∈V2lvx,vy∣G′+∑vx,vy∈V2lvx,vy∣G′.
Obviously,(26)∑vx,vy∈V1lvx,vy∣G=∑vx,vy∈V1lvx,vy∣G′,∑vx∈V1,vy∈V2lvx,vy∣G=+∑vx∈V1,vy∈V2lvx,vy∣G′,(27)lvx,vyG=lvx,vyG′=2,vx,vy∈Wvi,lvx,vyG>2=lvx,vyG′,vx∈Wvi,vy∈Wvj,i≠j.
Therefore,(28)ωG−ωG′=∑vx,vy∈V2lvx,vy∣G−∑vx,vy∈V2lvx,vy∣G′≥0,and the equality holds if and only if G≅G′.DenoteTn3=T16∪T22∪T23∪T24∪T25∪T26 (see Figure 13).By Lemma 6-7, we can verify that ifT∈Tn attains the minimum detour index of all graphs in Tn, then T^∈Tn3.Figure 13
GraphTn3.Remark 3.
In order to determine the tricyclic graphs which attain the minimum detour index of all graphs inTn, we just need to discuss the tricyclic graphs in T, where T^∈Tn3=T16,T22,T23,T24,T25,T26 (see Figure 13).


## 3. Results and Discussion

From the discussions of Section2, we can verify that if T∈Tn attains the minimum detour index of all graphs in Tn, then ωT=minωG, where G^∈Tn3=T16,T22,T23,T24,T25,T26.Theorem 1.
LetTn3 be defined as in Figure 13.(1)
Whenn=4 or n≥8, T26 is the unique graph which attains the minimum detour index of all graphs in Tn and ωT26=n2+4n−14.(2)
Whenn=5 or n=6, T25 is the unique graph which attains the minimum detour index of all graphs in Tn and ωT25=n2+5n−21.(3)
Whenn=7, T25 and T26 are the graphs which attain the minimum detour index of all graphs in Tn and ωT25=ωT26=63.Proof.
It can be checked directly that(29)Lv1∣T16=n+5;Lvi∣T16=3n−1,where2≤i≤7;Lw∣T16=2n+3,wherew∈Wv1.
Therefore,(30)ωT16=12∑u∈VT16Lu∣T16=12n+5+63n−1+2n+3n−7=n2+4n−11,n≥7.
Similarly, we have(31)ωT22=n2+5n−15=n2+4n+n−15,n≥6,ωT23=n2+8n−27=n2+4n+4n−27,n≥5,ωT24=n2+15n−57=n2+4n+11n−57,n≥6,ωT25=n2+5n−21=n2+4n+n−21,n≥5,ωT26=n2+4n−14,n≥4.(1)
Whenn=4 or n≥8, obviously, T26 is the unique graph which attains the minimum detour index of all graphs in Tn and ωT26=n2+4n−14.(2)
Whenn=5 or n=6, obviously, T25 is the unique graph which attains the minimum detour index of all graphs in Tn and ωT25=n2+5n−21.(3)
Whenn=7, obviously, T25 and T26 are the graphs which attain the minimum detour index of all graphs in Tn and ωT25=ωT26=63.


## 4. Conclusions

Mathematical chemistry is an area of research in chemistry in which mathematical tools are used to solve problems of chemistry. Chemical graph theory is an important area of research in mathematical chemistry which deals with topology of molecular structures such as the mathematical study of isomerism and the development of topological descriptors or indices. In this paper, we first introduce some useful graph transformations, and then we determine the minimum detour index of all tricyclic graphs. In addition, all the corresponding extremal graphs are characterized.



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*Source: 6031568-2019-10-13.xml*