Extremal conjugated unicyclic and bicyclic graphs with respect to total-eccentricity index

TL;DR

This paper investigates the extremal properties of conjugated unicyclic and bicyclic graphs concerning their total-eccentricity index, extending previous work on extremal trees and conjugated trees.

Contribution

It introduces new results characterizing extremal conjugated unicyclic and bicyclic graphs for the total-eccentricity index.

Findings

Identifies extremal conjugated unicyclic graphs with maximum and minimum total-eccentricity index.

Identifies extremal conjugated bicyclic graphs with maximum and minimum total-eccentricity index.

Extends known extremal graph results to conjugated unicyclic and bicyclic cases.

Abstract

Let $G$ be a molecular graph. The total-eccentricity index of graph $G$ is defined as the sum of eccentricities of all vertices of $G$. %In [R. Farooq, M.A. Malik, J. Rada, Extremal graphs with respect to total-eccentricity index, 2017, submitted], the authors studied The extremal trees, unicyclic and bicyclic graphs, and extremal conjugated trees with respect to total-eccentricity index are known. In this paper, we extend these results and study the extremal conjugated unicyclic and bicyclic graphs with respect to total-eccentricity index.

Equations

$$\mathrm{rad}(G)$$

$$\mathrm{diam}(G)$$

$$avec(G)=\frac{1}{n}\sum _{u\in V(G)}{e}_G(u).$$

$$\tau (G)=\sum _{u\in V(G)}{e}_G(u).$$

$$\tau (S_n)$$

$$\tau (C_n)$$

$$\tau (P_n)$$

$$\begin{array}[]{cc}\tau(\overline{U}_{1})=\frac{7}{2}n-3,&\tau(\overline{U}_{2})=\frac{3n^{2}}{4}-n-\frac{3}{4},\\[5.69046pt] \tau(\overline{B}_{1})=\frac{7}{2}n-4,&\tau(\overline{B}_{2})=\frac{3}{4}n^{2}-n-2.\end{array}$$

$$\tau (\overline{U})\ge 4n>\frac{7n}{2}-3=\tau ({\overline{U}}_1).$$

$$\tau (\overline{U})\ge 3(5)+4(n-5)=4n-5>\frac{7n}{2}-3=\tau ({\overline{U}}_1).$$

$$\tau (\overline{U})$$

$$\tau (\overline{B})\ge 4n>\frac{7n}{2}-4=\tau ({\overline{B}}_1).$$

$$\tau (\overline{B})\ge 3(7)+4(n-7)=4n-7>\frac{7n}{2}-4=\tau ({\overline{B}}_1).$$

$$\tau (\overline{B})$$

Full text

††Email addresses: alies.camp@gmail.com, mehar.ali@sns.nust.edu.pk (M. A. Malik), farook.ra@gmail.com (R. Farooq).

Keywords: Topological indices, total-eccentricity index, conjugated graphs.

AMS Classification: 05C05, 05C35

1 Introduction

Let $G$ be an $n$-vertex molecular graph with vertex set $V(G)$ and edge set $E(G)$. The vertices and edges of $G$ respectively correspond to atoms and chemical bonds between atoms. A topological index $T$ is a numerical quantity associated with the chemical structure of a molecule. The aim of such association is to correlate these indices with various physico-chemical properties of a chemical compound. An edge between two vertices $u,v\in V(G)$ is denoted by $uv$. The order and size of $G$ are respectively the cardinalities $|V(G)|$ and $|E(G)|$. The neighbourhood $N_{G}(v)$ of a vertex $v$ in $G$ is the set of vertices adjacent to $v$. A simple graph is a graph without loops and multiple edges. All graphs considered in this paper are simple. The degree $d_{G}(v)$ of a vertex $v$ in $G$ is the cardinality $|N_{G}(v)|$. A graph $G$ is called a $k$-regular graph if $d_{G}(v)=k$ for all $v\in V(G)$. A vertex of degree $1$ is called a pendant vertex. Let $P_{n}$, $C_{n}$ and $K_{n}$ respectively denote an $n$-vertex path, cycle and a complete graph. An $n$-vertex complete bipartite graph and $n$-vertex star is respectively denoted by $K_{a,b}$ and $K_{1,n-1}$ (or simply $S_{n}$), where $a+b=n$. A $(v_{1},v_{n})$-path with vertex set $\{v_{1},v_{2},\ldots,v_{n}\}$ is denoted by $v_{1}v_{2}\ldots v_{n}$. A graph $G$ is said to be connected if there exists a path between every pair of vertices in $G$. A maximal connected subgraph of a graph is called a component. A vertex $v\in V(G)$ is called a cut-vertex if deletion of $v$, along with the edges incident on it, increases the number of components of $G$. A maximal connected subgraph of a graph without any cut-vertex is called a block. A tree is a connected graph containing no cycle. Thus, an $n$-vertex tree is a connected graph with exactly $n-1$ edges. An $n$-vertex unicyclic graph is a simple connected graph which contains $n$ edges. Similarly, an $n$-vertex bicyclic graph is a simple connected graph which contains $n+1$ edges.

A matching $M$ in a graph $G$ is a subset of edges of $G$ such that no two edges in $M$ share a common vertex. A vertex $u$ in $G$ is said to be $M$-saturated if an edge of $M$ is incident with $u$. A matching $M$ is said to be perfect if every vertex in $G$ is $M$-saturated. A conjugated graph is a graph that contains a perfect matching. In graphs representing organic compounds, perfect matchings correspond to Kekulé structures, playing an important role in analysis of the resonance energy and stability of hydrocarbons [6]. The distance $d_{G}(u,v)$ between two vertices $u,v\in V(G)$ is defined as the length of a shortest path between $u$ and $v$ in $G$. If there is no path between vertices $u$ and $v$ then $d_{G}(u,v)$ is defined to be $\infty$. The eccentricity $e_{G}(v)$ of a vertex $v\in V(G)$ is defined as the largest distance from $v$ to any other vertex in $G$. The diameter ${\rm diam}(G)$ and radius ${\rm rad}(G)$ of a graph $G$ are respectively defined as:

[TABLE]

A vertex $v\in V(G)$ is said to be central if $e_{G}(v)={\rm rad}(G)$. The graph induced by all central vertices of $G$ is called the center of $G$, denoted as $C(G)$. A vertex $w$ is called an eccentric vertex of a vertex $v$ in $G$ if $d_{G}(v,w)=e_{G}(v)$. The set of all eccentric vertices of $v$ in a graph $G$ is denoted by $E_{G}(v)$.

The first topological index was introduced by Wiener [18] in 1947, to calculate the boiling points of paraffins. In 1971, Hosoya [8] defined the notion of Wiener index for any graph as the half sum of distances between all pairs of vertices. The average-eccentricity of an $n$-vertex graph $G$ was defined in 1988 by Skorobogatov and Dobrynin [16] as:

[TABLE]

In the recent literature, a minor modification of average-eccentricity index $avec(G)$ is used and referred as total-eccentricity index $\tau(G)$. It is defined as:

[TABLE]

The eccentric-connectivity index and the Randić index of a graph $G$ are defined respectively by $\xi(G)=\sum_{v\in V(G)}d_{G}(v)e_{G}(v)$ and $R(G)=\sum_{uv\in E(G)}(d_{G}(u)d_{G}(v))^{-\frac{1}{2}}$. Liang and Liu [11] proved a conjecture on the relation between the average-eccentricity and Randić index. Dankelmann and Mukwembi [3] obtained upper bounds on the average-eccentricity in terms of several graph parameters. Smith et al. [17] studied the extremal values of total-eccentricity index in trees. Ilic [9] studied some extremal graphs with respect to average-eccentricity. Farooq et al. [4] studied the extremal unicyclic and bicyclic graphs and extremal conjugated trees with respect to total-eccentricity index. For more details on topological indices of graphs and networks, the author is referred to [1, 2]. In this paper, we extend the results of [4] to conjugated unicyclic and bicyclic graphs.

For some special families of graphs of order $n\geq 4$, the total-eccentricity index is given as follows:

For a $k$-regular graph $G$, we have $\tau(G)=\frac{\xi(G)}{k}$, 2. 2.

$\tau(K_{n})=n$, 3. 3.

$\tau(K_{m,n})=2(m+n)$, $m,n\geq 2$, 4. 4.

The total-eccentricity index of a star $S_{n}$, a cycle $C_{n}$ and a path $P_{n}$ is given by

[TABLE]

Let $\{v_{1},v_{2},\ldots,v_{n}\}$ be the vertices of a path $P_{n}$. Let $U_{2}$ be a unicyclic graph obtained from $P_{n}$ by joining $v_{1}$ and $v_{3}$ by an edge. Similarly, let $B_{2}$ be a bicyclic graph obtained from $P_{n}$ by joining $v_{1}$ with two vertices $v_{3}$ and $v_{4}$. Note that when $n\equiv 0\pmod{2}$, the graphs $U_{2}$ and $B_{2}$ are conjugated and are denoted by $\overline{U}_{2}$ and $\overline{B}_{2}$, respectively. In Figure 1, we give two $7$-vertex bicyclic graphs $U_{2}$ and $B_{2}$. For $n\equiv 0\pmod{2}$, let $S_{n}^{*}$ be an $n$-vertex conjugated tree obtained by identifying one vertex each from $\frac{n-2}{2}$ copies of $P_{3}$ and deleting a single pendent vertex. Let $v$ be the unique central vertex of $S_{n}^{*}$. Let $\overline{U}_{1}$ be a conjugated unicyclic graph obtained from $S_{n}^{*}$ by adding an edge between $v$ and any vertex not adjacent to $v$. In a similar fashion, let $\overline{B}_{1}$ be a conjugated bicyclic graph obtained from $S_{n}^{*}$ by adding two edges between $v$ and any two vertices not adjacent to $v$ (see Figure 2).

Now we give some previously known results on the center of a graph from [7] and some results on extremal graphs with respect to total-eccentricity index from [4]. In next theorem, we give a result dealing with the location of center in a connected graph.

Theorem 1.1** (Harary and Norman [7]).**

The center of a connected graph $G$ is contained in a block of $G$.

The only possible blocks in a unicyclic graph are $K_{1}$, $K_{2}$ or a cycle $C_{k}$. Thus the following corollary gives the center of an $n$-vertex conjugated unicyclic graph $\overline{U}$.

Corollary 1.2**.**

If $\overline{U}$ is an $n$-vertex conjugated unicyclic graph with a unique cycle $C_{k}$, then $C(\overline{U})=K_{1}$ or $K_{2}$, or $C(\overline{U})\subseteq C_{k}$

The following results give the extremal unicyclic and bicyclic graphs with respect to total-eccentricity index.

Theorem 1.3** (Farooq et al. [4]).**

Among all $n$-vertex unicyclic graphs, $n\geq 4$, the graph $U_{2}$ shown in Figure 1 has maximal total-eccentricity index.

Theorem 1.4** (Farooq et al.[4]).**

Among all $n$-vertex bicyclic graphs, $n\geq 5$, the graph $B_{2}$ shown in Figure 1 has the maximal total-eccentricity index.

In Section 2, we find extremal conjugated unicyclic and bicyclic graphs with respect to total-eccentricity index.

2 Conjugated unicyclic and bicyclic graphs

In this section, we find extremal conjugated unicyclic and bicyclic graphs with respect to total-eccentricity index. In (2.1), we give the total-eccentricity index of the conjugated graphs $\overline{U}_{1}$, $\overline{U}_{2}$, $\overline{B}_{1}$ and $\overline{B}_{2}$ which can easily be computed.

[TABLE]

Using Theorem 1.1 and Corollary 1.2, we prove the following result.

Remark 2.1**.**

When $n=4$, the graph shown in Figure 3(a) has the smallest total-eccentricity index among all $4$-vertex conjugated unicyclic graphs. When $n=6$, the graphs shown in Figure 3(b) and Figure 3(c) have smallest total-eccentricity index among $6$-vertex conjugated unicyclic graphs. When $n=8$, the graph shown in Figure 3(d) has smallest total-eccentricity index among $8$-vertex conjugated unicyclic graphs.

Theorem 2.1**.**

Let $n\equiv 0\pmod{2}$ and $n\geq 10$. Then among all $n$-vertex conjugated unicyclic graphs, the graph $\overline{U}_{1}$ shown in Figure 2 has the minimal total-eccentricity index.

Proof.

Let $\overline{U}_{1}$ be the $n$-vertex conjugated unicyclic graph shown in Figure 2. Let $\overline{U}$ be an arbitrary $n$-vertex conjugated unicyclic graph with a unique cycle $C_{k}$. We show that $\tau(\overline{U})\geq\tau(\overline{U_{1}})$. Let $n_{i}$ denote the number of vertices with eccentricity $i$ in $\overline{U}$. If $k\geq 8$ or ${\rm rad}(\overline{U})\geq 4$ then

[TABLE]

In the rest of the proof, we assume that $k\in\{3,4,5,6,7\}$ and ${\rm rad}(\overline{U})\in\{2,3\}$. Let $k\in\{6,7\}$. If $x$ is a vertex of $\overline{U}$ such that $x$ is not on $C_{k}$, then $e_{\overline{U}}(x)\geq 4$. Also, it is easily seen that there are at most five vertices on $C_{k}$ with eccentricity $3$. Thus

[TABLE]

We complete the proof by considering the following cases.

Case 1. When ${\rm rad}(\overline{U})=3$ and $k\in\{3,4,5\}$. By Corollary 1.2, $C(\overline{U})=K_{1}$ or $C(\overline{U})=K_{2}$ or $C(\overline{U})\subseteq C_{k}$. This shows that $\overline{U}$ has at most five vertices with eccentricity $3$. Thus the inequality (2.2) holds in this case.

Case 2. When ${\rm rad}(\overline{U})=2$ and $k\in\{4,5\}$. Then ${\rm diam}(\overline{U})\leq 2{\rm rad}(\overline{U})=4$ and there will be exactly one vertex in $C(\overline{U})$. That is, $n_{2}=1$. Let $v$ be the vertex with $e_{\overline{U}}(v)=2$. Then $v\in V(C_{k})$. Considering several possibilities for longest possible paths (of length $2$) starting from $v$ and that $\overline{U}$ is conjugated, one can see that $\overline{U}$ is isomorphic to one of the graphs shown in Figure 4. Moreover, observe that $n_{4}\geq\frac{n}{2}-1$.

Since $n_{2}+n_{3}+n_{4}=n$ and $n_{2}=1$, we can write

[TABLE]

Case 3. When ${\rm rad}(\overline{U})=2$ and $k=3$. Then $n_{2}=1$ (see Figure 5). Let $v$ be the unique central vertex of $\overline{U}$. Then either $v$ is a vertex of $C_{3}$ or $v$ is adjacent to a vertex of $C_{3}$. When $v\in V(C_{3})$, then $\overline{U}$ is isomorphic to one of the graphs shown in Figure 5(a), 5(b) or 5(c). In this case, all vertices with eccentricity $4$ are pendent. This gives $n_{4}\geq\frac{n}{2}-2$. Therefore

[TABLE]

Similarly, if the central vertex $v$ is not on $C_{3}$, then $\overline{U}$ is isomorphic to one of the graphs shown in Figure 5(d) or 5(e). Note that $n_{4}\geq\frac{n}{2}$. Thus $\tau(\overline{U})\geq 2(1)+3(n-1)+\frac{n}{2}=\frac{7n}{2}-1>\tau(\overline{U}_{1}).$

Combining all the cases, we see that $\overline{U}_{1}$ is the minimal graph with respect to total-eccentricity index. This completes the proof.

∎

The following theorem gives the maximal conjugated unicyclic graphs with respect to total-eccentricity index.

Theorem 2.2**.**

Let $n\equiv 0(\bmod 2)$. Then the $n$-vertex conjugated unicyclic graph corresponding to the maximal total-eccentricity index is the graph $\overline{U}_{2}$ shown in Figure 2.

Proof.

Note that the class of all $n$-vertex conjugated unicyclic graphs forms a subclass of the class of all $n$-vertex unicyclic graphs. From Theorem 1.3 we see that among all $n$-vertex unicyclic graphs, the graph $U_{2}$ (see Figure 1) has the largest total-eccentricity index. Since $U_{2}$ admits a a perfect matching when $n\equiv 0(\bmod 2)$, the result follows. ∎

Corollary 2.3**.**

For an $n$-vertex conjugated unicyclic graph $\overline{U}$, we have $\frac{7n}{2}-3\leq\tau(\overline{U})\leq\frac{3n^{2}}{4}-n-\frac{3}{4}$.

Proof.

Using Theorem 2.1, Theorem 2.2 and equation (2.1), we obtain the required result. ∎

The next theorem gives the minimal conjugated bicyclic graphs with respect to total-eccentricity index.

Remark 2.2**.**

Let $n=4$. Then among all $4$-vertex conjugated bicyclic graphs, one can easily see that the graph shown in Figure 6(a) has the minimal total-eccentricity index. Similarly, when $n=6$ and $8$, then the graphs respectively shown in Figure 6(b) and Figure 6(c) have the minimal total-eccentricity index among all $6$-vertex and $8$-vertex conjugated bicyclic graphs.

Theorem 2.4**.**

Let $n\equiv 0\pmod{2}$ and $n\geq 10$. Then among the $n$-vertex conjugated bicyclic graphs, the graph $\overline{B}_{1}$ shown in Figure 2 has the minimal total-eccentricity index.

Proof.

Let $\overline{B}_{1}$ be the $n$-vertex conjugated bicyclic graph shown in Figure 2. Let $\overline{B}$ be an arbitrary $n$-vertex conjugated bicyclic graph and $\overline{B}\ncong\overline{B}_{1}$. Let $C(\overline{B})$ denote the center of $\overline{B}$ and $n_{i}$ denote the number of vertices with eccentricity $i$. The proof is divided into two cases depending upon the number of cycles in $\overline{B}$.

Case 1. When $\overline{B}$ contains two edge-disjoint cycles $C_{k_{1}}$ and $C_{k_{2}}$ of lengths $k_{1}$ and $k_{2}$, respectively. Without loss of generality, assume that $k_{1}\leq k_{2}$. If ${\rm rad}(\overline{B})\geq 4$ or $k_{2}\geq 8$, then

[TABLE]

Thus, we assume that $k_{2}\in\{3,4,5,6,7\}$ and ${\rm rad}(\overline{B})\in\{2,3\}$. If $k_{2}\in\{6,7\}$, then for any vertex $x\notin V(C_{k_{2}})$, $e_{\overline{B}}(x)\geq 4$. Moreover, as $k_{2}\leq 7$, the number of vertices with eccentricity $3$ are at most $7$. Thus

[TABLE]

We consider the following three subcases.

(a) Let ${\rm rad}(\overline{B})=3$ and $k_{2}\in\{3,4,5\}$. By Theorem 1.1, we have $|V(C(\overline{B}))|\leq 7$. Thus $\tau(\overline{B})$ satisfies equation (2.3).

(b) Let ${\rm rad}(\overline{B})=2$ and $k_{2}\in\{4,5\}$. Take $v\in V(C(\overline{B}))$. We observe that the center $C(\overline{B})$ is contained in $C_{k_{2}}$ and $n_{2}=1$. Assume $v$ to be the unique central vertex of $\overline{B}$. Then for several possible choices for possible pendent vertices in the conjugated graph $\overline{B}$, one can observe that $\overline{B}$ is one of the graphs shown in Figure 7. Moreover $n_{4}\geq\frac{n}{2}-2$. Then $\tau(\overline{B})\geq 2n_{2}+3n_{3}+4n_{4}=2n_{2}+3(n_{3}+n_{4})+n_{4}\geq 2(1)+3(n-1)+\frac{n}{2}-2=\frac{7n}{2}-3>\tau(\overline{B}_{1})$.

(c) Let ${\rm rad}(\overline{B})=2$ and $k_{1}=k_{2}=3$. If $C(\overline{B})\subseteq C_{k_{2}}$ then $C(\overline{B})=K_{1}$, otherwise $n\leq 8$ which is not true. Similarly, if $C(\overline{B})\nsubseteq C_{k_{2}}$, then $C(\overline{B})=K_{2}$ or $K_{1}$. Since $n_{2}\ngeq 2$, we have $n_{2}=1$. Let $c$ be the unique central vertex. Then $\overline{B}$ is isomorphic to one of the graphs shown in Figure 8. When $c$ is not a vertex of $C_{k_{1}}$ or $C_{k_{2}}$, then $n_{4}\geq\frac{n}{2}+1$ and $\tau(\overline{B})\geq 2(1)+3(n-1)+\frac{n}{2}+1=\frac{7n}{2}>\tau(\overline{B}_{1})$. If $c$ is a vertex of $C_{k_{1}}$ or $C_{k_{2}}$, then $n_{4}\geq\frac{n}{2}-3$. Thus $\tau(\overline{B})\geq 2(1)+3(n-1)+\frac{n}{2}-3=\frac{7n}{2}-4=\tau(\overline{B}_{1})$.

Case 2. When cycles of $\overline{B}$ share some edges. Then there are cycles $C_{k_{1}}$, $C_{k_{2}}$ and $C_{k_{3}}$ in $\overline{B}$ of lengths $k_{1}$, $k_{2}$ and $k_{3}$, respectively. Without loss of generality, assume that $k_{1}\leq k_{2}\leq k_{3}$. Then $k_{1},k_{2}\geq 3$ and $k_{3}\geq 4$. Let $Q$ be the subgraph of $\overline{B}$ induced by the vertices of $C_{k_{1}}$, $C_{k_{2}}$ and $C_{k_{3}}$. Clearly, $Q$ contains the cycle $C_{k_{3}}$. Assume that $V(C_{k_{3}})=\{v_{1},v_{2},\ldots,v_{k_{3}}\}$. Then $\overline{B}$ is minimal with respect to total-eccentricity index if $Q$ can be obtained from the cycle $C_{k_{3}}$ by adding the edge $v_{1}v_{\lfloor\frac{k_{3}}{2}\rfloor+1}$ or $v_{1}v_{\lfloor\frac{k_{3}}{2}\rfloor+2}$.

When $k_{3}\geq 13$, then $e_{\overline{B}}(w)\geq 4$ for all $w\in V(\overline{B})$. This gives us $\tau(\overline{B})\geq 4n>\tau(\overline{B}_{1})$. Assume that ${\rm rad}(\overline{B})\leq 3$ and $k_{3}\leq 12$. We consider the following two subcases:

(a) When $k_{3}\in\{9,10,11,12\}$. Then ${\rm rad}(\overline{B})=3$. There exists a vertex $c$ with degree $3$ in $Q$ such that $c\in C(\overline{B})$ and $c$ has at most $\frac{n-10}{2}+1$ neighbours not in $Q$. Clearly, all of these vertices will have eccentricity $4$. Then $\frac{n}{2}-5$ such vertices can have unique pendent vertices with eccentricity $5$. Moreover, there will be at least one vertex with eccentricity $5$ in $Q$, otherwise ${\rm rad}(\overline{B})\neq 3$. Thus $n_{5}\geq\frac{n}{2}-4$. Moreover, at most $3$ vertices in $Q$ can have eccentricity $3$. This gives $1\leq n_{3}\leq 3$. When $k_{3}=9$, such a graph $\overline{B}$ is shown in Figure 9(a). The vertex $v\in V(Q)$ such that $e_{\overline{B}}(v)=5$ is also shown in the figure. Using the facts that $n=n_{3}+n_{4}+n_{5}$ and $n_{4}+n_{5}=n-n_{3}\geq n-3$. We get

[TABLE]

(b) When $k_{3}\in\{4,5,6,7,8\}$. We first assume that ${\rm rad}(\overline{B})=3$. Then by Theorem 1.1, we have $1\leq n_{3}\leq 8$. Thus $\tau(\overline{B})\geq 3(8)+4(n-8)=4n-8\geq\tau(\overline{B}_{1})$. On the other hand, when ${\rm rad}(\overline{B})=2$. Then $n_{2}=1$. When $C(\overline{B})\nsubseteq Q$, then the minimal graph $\overline{B}$ is isomorphic to the graph shown in Figure 9(b). Clearly, $n_{4}\geq\frac{n}{2}$. Thus $\tau(\overline{B})=2n_{2}+3(n_{3}+n_{4})+n_{4}\geq 2(1)+3(n-1)+\frac{n}{2}=\frac{7n}{2}-1>\tau(\overline{B}_{1})$. When $C(\overline{B})\subseteq Q$, then $\overline{B}$ is isomorphic to one of the graphs shown in Figures 9(c)$-$9(h). It can be seen that $n_{4}\geq\frac{n}{2}-2$. Thus we can write $\tau(\overline{B})=2n_{2}+3(n_{3}+n_{4})+n_{4}\geq 2(1)+3(n-1)+\frac{n}{2}-2=\frac{7n}{2}-3=\tau(\overline{B}_{1})$.

Combining the results of Case 1 and Case 2, we see that among all conjugated bicyclic graphs, $\overline{B}_{1}$ has the minimal total-eccentricity index. The proof is complete. ∎

Theorem 2.5**.**

Let $n\equiv 0\pmod{2}$. Then among the $n$-vertex conjugated bicyclic graphs, the graph $\overline{B}_{2}$ shown in Figure 2 has the maximal total-eccentricity index.

Proof.

For $n\equiv 0(\bmod 2)$, the proof can be derived from the proof of Theorem 1.4. ∎

Corollary 2.6**.**

For any conjugated bicyclic graph $\overline{B}$, we have $\frac{7n}{2}-4\leq\tau(\overline{B})\leq\frac{3n^{2}}{4}-n-2$.

Proof.

The result follows by using Theorem 2.4, Theorem 2.5 and equation (2.3). ∎

3 Conclusion

In this paper, we extended the results of Farooq et al. [4] and studied the extremal conjugated unicyclic and bicyclic graphs with respect to total-eccentricity index.

Acknowledgements

The authors are thankful to the Higher Education Commission of Pakistan for supporting this research under the grant 20-3067/NRPU/R$\&$D/HEC/12/831.