# The Average Eccentricity of Block Graphs: A Block Order Sequence Perspective

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## Abstract

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## 1. Introduction

## 2. Preliminaries

**Lemma 1.**

**Proof.**

- u and v are in A and C, respectively.
- There is a vertex $w\in V\left(f\right(u\left)\right)$ such that w and v are adjacent in the graph G.

**Theorem 1.**

- The number of blocks in G is$$\alpha =\underset{i=1}{\overset{k}{\Sigma}}{s}_{i}$$
- The order of G is$$n=\underset{i=1}{\overset{k}{\Sigma}}\left({b}_{i}{s}_{i}\right)-\alpha +1=\underset{i=1}{\overset{k}{\Sigma}}\left({s}_{i}({b}_{i}-1)\right)+1$$
- The size of G is$$m=\frac{1}{2}\underset{i=1}{\overset{k}{\Sigma}}\left(({b}_{i}^{2}-{b}_{i}){s}_{i}\right)$$

**Proof.**

**Theorem 2.**

**Proof.**

**The lower and upper bounds of the average eccentricity on block graphs**. We established an equivalence relation on the set of graphs with order n from the perspective of block order sequence, which is going to be presented in Theorem 3. The equivalence relation naturally partitions the set of block graphs with order n into several equivalent classes. Recall that**all graphs in every such equivalent class have the same block order sequence**. Thus, to bind the average eccentricity on block graphs with order, n is transformed to bind the value on every equivalent class. This transformation seems independently interesting.**A linear time algorithm to find out a block order sequence**. Algorithm 1 is devised to find out the block order sequence of a block graph. The algorithm is proven to be in linear time by Theorem 16. This result shows that it is practicable and available to study the eccentricity on block graphs from the perspective of its block order sequence.

## 3. Extremal Values on Block Graphs with Order n

**Theorem 3.**

**Proof.**

- For every graph $G\in \mathcal{G}\left(n\right)$, $(G,G)\in \mathcal{R}$, so it is reflexive.
- $(G,H)\in \mathcal{R}$ if and only if $(H,G)\in \mathcal{R}$ for every two graphs $G,H\in \mathcal{G}\left(n\right)$, so it is symmetric.
- for ${G}_{1},{G}_{2},{G}_{3}\in \mathcal{G}\left(n\right)$, if $({G}_{1},{G}_{2})\in \mathcal{R}$ and $({G}_{2},{G}_{3})\in \mathcal{R}$, then $({G}_{1},{G}_{3})$ must be in the relation $\mathcal{R}$, so it is transitive.

**Theorem 4.**

**Theorem 5.**

## 4. Bounds on Block Graphs with a Fixed Block Order Sequence

#### 4.1. The Lower Bound and Corresponding Extremal Graphs

**Block-slide transformation**: Let ${B}_{i}$ be a block in a block graph G depicted in Figure 2, where u and v are two distinct cut-vertices of ${B}_{i}$. Let $A={G}_{(u,v)}$ and $B={G}_{(v,u)}$ be two subgraphs in G (see Figure 2). Let ${G}^{\prime}=G-\{(u,w)\in E\left(G\right):w\in {N}_{A}\left(u\right)\}+\{(v,w):w\in {N}_{A}\left(u\right)\}$. The transformation from G to ${G}^{\prime}$ is named as a block-slide transformation on G. On the other side, the transformation from ${G}^{\prime}$ to G is named as an inverse block-slide transformation on ${G}^{\prime}$, i.e., $G={G}^{\u2033}-\{(v,w)\in E\left({G}^{\prime}\right):w\in {N}_{A}\left(v\right)\}+\{(u,w):w\in {N}_{A}\left(v\right)\}$.

**Theorem 6.**

**Theorem 7.**

**Proof.**

**Case 1**: $\u03f5({u}_{1},{A}_{1})\le \u03f5({u}_{{b}_{i}},{A}_{{b}_{i}})\le L$. For every vertex $v\in ({\displaystyle \bigcup _{2\le j\le {b}_{i}-1}}V\left({A}_{j}\right))\cup V\left({B}_{i}\right)\setminus \left\{{u}_{{b}_{i}}\right\}$, we have $\u03f5(v,G)=\u03f5(v,{G}^{\u2033})$. For every vertex $v\in V\left({A}_{1}\right)\setminus \left\{{u}_{1}\right\}$, $\u03f5(v,G)\ge \u03f5(v,{G}^{\u2033})$. For every vertex $v\in V\left({A}_{{b}_{i}}\right)$, $\u03f5(v,G)\ge \u03f5(v,{G}^{\u2033})$. Hence, the average eccentricity of the whole graph does not increase after a block-slide transformation.

**Case 2**: $\u03f5({u}_{{b}_{i}},{A}_{{b}_{i}})<\u03f5({u}_{1},{A}_{1})<L$. It is easy to verify that the eccentricity of every vertex in G does not change under the transformation. So the average eccentricity of the whole graph remains the same after a block-slide transformation.

**Case 3**: $\u03f5({u}_{{b}_{i}},{A}_{{b}_{i}})<\u03f5({u}_{1},{A}_{1})=L$. It is easy to verify that if there is a subgraph ${A}_{j}:2\le j\le {b}_{i}-1$ such that $\u03f5({u}_{j},{A}_{j})=L$, then the average eccentricity of the whole graph remains the same after the transformation. Let us consider the case that there is no integer $j:2\le j\le {b}_{i}-1$ such that $\u03f5({u}_{j},{A}_{j})=L$.

- $\u03f5(v,G)=\u03f5(v,{G}^{\prime})$ holds for every vertex $v\in V\left(G\right)\setminus V\left({A}_{1}\right)\setminus V\left({A}_{{b}_{i}}\right)$.
- $\u03f5(v,G)\ge \u03f5(v,{G}^{\prime})$ holds for every vertex $v\in V\left({A}_{1}\right)\setminus \left\{{u}_{1}\right\}$.
- $\u03f5(v,G)>\u03f5(v,{G}^{\prime})$ holds for every vertex $v\in V\left({A}_{{b}_{i}}\right)$.
- $\u03f5({u}_{1},G)-\u03f5({u}_{1},{G}^{\prime})=1$.

**Corollary 1.**

**Theorem 8.**

**Proof.**

#### 4.2. The Upper Bound and Corresponding Extremal Graphs

**Block-shift transformation**: Let ${B}_{i}$ be a block in a block graph G where t, v, and w are all cut-vertices in ${B}_{i}$ as depicted in Figure 4. Note that the three cut-vertices t, w, and v do not need to be distinct to each other. Let C be a petal at the vertex t such that C is a path-like block graph and does not contain the block ${B}_{i}$. Let $u\in V\left(C\right)$ be a pendent vertex of G. Let A and B be two sets of petals corresponding, respectively, to vertices w and v, such that $\u03f5(w,A)\le \u03f5(v,B)$ holds. Let ${G}^{\prime}=G-\{(w,s)\in E\left(G\right):s\in {N}_{A}\left(w\right)\}+\{(u,s):s\in {N}_{A}\left(w\right)\}$. Then the transformation from G to ${G}^{\prime}$ is said to be a block-shift transformation on graph G, while the transformation from ${G}^{\prime}$ to G is said to be an inverse block-shift transformation on graph ${G}^{\prime}$, i.e., $G={G}^{\prime}-\{(u,s)\in E\left({G}^{\prime}\right):s\in {N}_{A}\left(u\right)\}+\{(w,s):s\in {N}_{A}\left(u\right)\}$.

**Theorem 9.**

**Proof.**

**Corollary 2.**

**Theorem 10.**

## 5. Bounds and Extremal Graphs for Path-like Block Graphs

#### 5.1. Formulas for the Average Eccentricity on Path-like Block Graphs with Given Block Order Sequence

**Theorem 11.**

- If the number of blocks α is even, then$$aecc\left(G\right)=\frac{1}{n}(\beta -\frac{3}{4}{\alpha}^{2})$$

**Proof.**

#### 5.2. Extremal Graphs for Path-like Block Graphs with Given Block Order Sequence

- If $\alpha $ is even, then$$M\left(i\right)=\left\{\begin{array}{c}i\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}1\le i\le \frac{\alpha}{2}\hfill \\ \frac{\alpha}{2}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}i=\frac{\alpha}{2}+1\hfill \\ M(i-1)-1\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\frac{\alpha}{2}+2\le i\le \alpha \hfill \end{array}\right.$$
- If $\alpha $ is odd, then$$M\left(i\right)=\left\{\begin{array}{c}i\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}1\le i\le \lceil \frac{\alpha}{2}\rceil \hfill \\ M(i-1)-1\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\lceil \frac{\alpha}{2}\rceil +1\le i\le \alpha \hfill \end{array}\right.$$

**Lemma 2.**

**Proof.**

**Case 1**: $1\le i\le \frac{\alpha}{2}$. The eccentricity of every vertex $u\in V\left({B}_{i}\right)\setminus \left\{{v}_{i}\right\}$ is $\alpha -i+1=\alpha -M\left(i\right)+1$, while the vertex ${v}_{i}$ has its eccentricity equal to the value of $\alpha -i$. Hence, the lemma holds on this case.

**Case 2**: $i=\frac{\alpha}{2}+1$. The eccentricity of every vertex $u\in V\left({B}_{i}\right)\setminus \left\{{v}_{i-1}\right\}$ is equal to $\frac{\alpha}{2}+1=\alpha -\frac{\alpha}{2}+1=\alpha -M\left(i\right)+1$, while the vertex ${v}_{i-1}$ has the eccentricity as $\frac{\alpha}{2}$. Hence, the lemma holds on this case.

**Case 3**: $\frac{\alpha}{2}+2\le i\le \alpha $. The eccentricity of every vertex $u\in V\left({B}_{i}\right)\setminus \left\{{v}_{i-1}\right\}$ is $\alpha -M\left(i\right)+1$, while the vertex ${v}_{i-1}$ has the value of $\alpha -M\left(i\right)$. Hence, the lemma holds on this case.

**Block-exchange transformation**: Let G be a path-like block graph with labeled blocks and cut-vertices as depicted in Figure 5. If there are two distinct blocks ${B}_{i}$, ${B}_{j}$ ($1\le i,j\le \alpha $) such that both ${N}_{i}<{N}_{j}$ and $M\left(i\right)<M\left(j\right)$ hold, then we construct a new graph ${G}^{\prime}$ by exchanging the two blocks ${B}_{i}$ and ${B}_{j}$ in G, i.e., ${G}^{\prime}=G-\{({v}_{i-1},u):u\in {N}_{{B}_{i}}\left({v}_{i-1}\right)\}-\{({v}_{i},u):u\in {N}_{{B}_{i}}\left({v}_{i}\right)\}+\{({v}_{j-1},u):u\in {N}_{{B}_{i}}\left({v}_{i-1}\right)\}+\{({v}_{j},u):u\in {N}_{{B}_{i}}\left({v}_{i}\right)\}-\{({v}_{j-1},u):u\in {N}_{{B}_{j}}\left({v}_{j-1}\right)\}-\{({v}_{j},u):u\in {N}_{{B}_{j}}\left({v}_{j}\right)\}+\{({v}_{i-1},u):u\in {N}_{{B}_{j}}\left({v}_{j-1}\right)\}+\{({v}_{i},u):u\in {N}_{{B}_{j}}\left({v}_{j}\right)\}$ (see Figure 6). The transformation from G to ${G}^{\prime}$ is called a block-exchange transformation on G. On the other side, the transformation from ${G}^{\prime}$ to G is said to be an inverse block-exchange transformation on ${G}^{\prime}$. Note that, after the block-exchange transformation, ${B}_{i}^{{}^{\prime}}\cong {B}_{j}$ and ${B}_{j}^{{}^{\prime}}\cong {B}_{i}$ holds in graph ${G}^{\prime}$.

**Theorem 12.**

**Proof.**

**Corollary 3.**

**Theorem 13.**

**Proof.**

#### 5.3. Bounds on Path-like Graphs with Order n

**Theorem 14.**

**Proof.**

## 6. Extracting the Block Order Sequence of a Block Graph

#### 6.1. To Decide a Cut-Vertex

**Lemma 3.**

**Proof.**

**Theorem 15.**

**Proof.**

#### 6.2. To Obtain a Block Order Sequence

**Lemma 4.**

**Proof.**

Algorithm 1: Seq(G, v, visit, B) |

Algorithm 2: Is_CutVertex(G, v, u) |

**Theorem 16.**

**Proof.**

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Buckley, F.; Harary, F. Distance in Graphs; Addisson-Wesley: Redwood City, CA, USA, 1990. [Google Scholar]
- Cormen, T.H.; Leiserson, C.E.; Rivest, R.L.; Stein, C. Introduction to Algorithme; MIT Press: Cambridge, UK, 2001. [Google Scholar]
- Ali, P.; Dankelmann, P.; Morgan, M.J.; Mukwembi, S.; Vetrik, T. The average eccentricity, spanning trees of plane graphs, size and order. Util. Math.
**2018**, 107, 37–49. [Google Scholar] - Dankelmann, P.; Goddard, W.; Swart, C.S. The average eccentricity of a graph and its subgraphs. Util. Math.
**2004**, 41, 41–51. [Google Scholar] - Dankelmann, P.; Mukwembi, S. Upper bounds on the average eccentricity. Discret. Appl. Math.
**2014**, 167, 72–79. [Google Scholar] [CrossRef] - Dankelmann, P.; Osaye, F.J. Average eccentricity, k-packing and k-domination in graphs. Discret. Math.
**2019**, 342, 1261–1274. [Google Scholar] [CrossRef] - Dankelmann, P.; Osaye, F.J.; Mukwembi, S.; Rodrigues, B. Upper Bounds on the average eccentricity of K3-free and C4-free graphs. Discret. Appl. Math. 2019; in press. [Google Scholar] [CrossRef]
- Du, Z.; Ilić, A. On AGX conjectures regarding average eccentricity. MATCH Commun. Math. Comput. Chem.
**2013**, 69, 597–609. [Google Scholar] - Du, Z.; Ilić, A. A proof of the conjecture regarding the sum of the domination number and average eccentricity. Discret. Appl. Math.
**2016**, 201, 105–113. [Google Scholar] [CrossRef] - Ilić, A. On the extremal properties of the average eccentricity. Comput. Math. Appl.
**2012**, 64, 2877–2885. [Google Scholar] [CrossRef] [Green Version] - Smith, H.; Székely, L.A.; Wang, H. Eccentricity sum in trees. Discret. Appl. Math.
**2016**, 207, 120–131. [Google Scholar] [CrossRef] [Green Version] - Tang, Y.; Zhou, B. On average eccentricity. MATCH Commun. Math. Comput. Chem.
**2012**, 67, 405–423. [Google Scholar] - Ghorbani, M.; Li, X.; Maimani, H.R.; Mao, Y.; Rahmani, S.; Rajabi-Parsa, M. Steiner (revised) Szeged index of graphs. MATCH Commun. Math. Comput. Chem.
**2019**, 82, 733–742. [Google Scholar] - Lei, H.; Shi, Y.; Yue, J. A Survey on the Wiener Polarity Index. MATCH Commun. Math. Comput. Chem.
**2021**, 86, 289–318. [Google Scholar] - Yu, G.; Feng, L. On the connective eccentricity index of graphs. MATCH Commun. Math. Comput. Chem.
**2013**, 69, 611–628. [Google Scholar] - Zhou, B.; Du, Z. On eccentric connectivity index. MATCH Commun. Math. Comput. Chem.
**2010**, 63, 181–198. [Google Scholar] - Mao, Y.; Furtulab, B. Steiner Distance in Chemical Graph Theory. MATCH Commun. Math. Comput. Chem.
**2021**, 86, 211–287. [Google Scholar] - Zaheer Ahmad, A.Q.; Baig, M.; Azhar, M. Imran Total eccentricity and average eccentricity indices of some chemical graphs: Carbon graphite and copper oxide. J. Inf. Optim. Sci.
**2020**, 41, 905–924. [Google Scholar] - Bondy, J.A.; Murty, U.S.R. Graph theory with applications. In Great Britain; The Macmillan Press: Stuttgart, Germany, 1976. [Google Scholar]
- Goddard, W.; Rosen, K.H.; Shier, D.R. Handbook of Discrete and Combinatorial Mathematics, 2nd ed.; Chapman and Hall: London, UK; CRC Press: Boca Raton, FL, USA, 2018. [Google Scholar]
- Jäntschi, L.; Diudea, M. Subgraphs of pair vertices. J. Math. Chem.
**2019**, 45, 364–371. [Google Scholar] [CrossRef] - Jäntschi, L.; Balan, M.C.; Bolboacă, S. Counting Polynomials on Regular Iterative Structures. Appl. Med. Inform.
**2009**, 24, 67–95. [Google Scholar] - Diudea, M.V.; Gutman, I.; Jantschi, L. Molecular Topology; Nova Science Publishers: New York, NY, USA, 2001. [Google Scholar]

**Figure 1.**The kinds of neighbors of v and the subgraphs ${G}_{({v}_{i},v)}:i=1,2,3$ in a block graph G.

**Figure 2.**${G}^{\prime}$ is obtained by a block-slide transformation from G, where u and v are on the same block ${B}_{i}$ and are two distinct cut-vertices of ${B}_{i}$.

**Figure 4.**${G}^{\prime}$ is obtained by a block-shift transformation from G where v, t, and w are all cut-vertices on the same block ${B}_{i}$. Note that the three cut-vertices v, t, and w are not restricted to be distinct. The longest path which ends with w in the subgraph A is not longer than the longest path which ends with v in the subgraph B. Moreover, the subgraph C is a path-like block subgraph.

**Figure 5.**A path-like block graph G is straightened on a horizontal line from left to right and labeled ${B}_{1}$, ${B}_{2}$, ${B}_{3}$,…,${B}_{\alpha}$ one by one, where $\alpha $ is the number of blocks in G.

**Figure 6.**${G}^{\prime}$ is obtained by a block-exchange transformation from G, where ${B}_{i}^{{}^{\prime}}\cong {B}_{j}$, ${B}_{j}^{{}^{\prime}}\cong {B}_{i}$, and ${B}_{s}^{{}^{\prime}}\cong {B}_{s}$ for $s\ne i,s\ne j$. Moreover, ${v}_{s}={v}_{s}^{{}^{\prime}}$ for every $1\le s\le \alpha -1$.

**Figure 7.**A path-like block graph G is straightened on a horizontal line from left to right and labeled ${B}_{1}$, ${B}_{2}$, ${B}_{3}$,…,${B}_{\alpha}$ one by one, where $\alpha $ is the number of blocks in G, where the two non-cut-vertices ${v}_{0}$ and ${v}_{\alpha}$ are labeled ${B}_{0}$ and ${B}_{\alpha +1}$, respectively.

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**MDPI and ACS Style**

Li, X.; Yu, G.; Das, K.C.
The Average Eccentricity of Block Graphs: A Block Order Sequence Perspective. *Axioms* **2022**, *11*, 114.
https://doi.org/10.3390/axioms11030114

**AMA Style**

Li X, Yu G, Das KC.
The Average Eccentricity of Block Graphs: A Block Order Sequence Perspective. *Axioms*. 2022; 11(3):114.
https://doi.org/10.3390/axioms11030114

**Chicago/Turabian Style**

Li, Xingfu, Guihai Yu, and Kinkar Chandra Das.
2022. "The Average Eccentricity of Block Graphs: A Block Order Sequence Perspective" *Axioms* 11, no. 3: 114.
https://doi.org/10.3390/axioms11030114