# Ad-Hoc Lanzhou Index

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## Abstract

**:**

## 1. Introduction

## 2. Identities

**Observation**

**1.**

**Proposition**

**1.**

**Proof.**

**Proposition**

**2.**

**Proof.**

## 3. Extremal Results Concerning $\mathit{\xi}$-Cyclic Graphs

**Lemma**

**1.**

**Proof.**

**Lemma**

**2.**

**Proof.**

**Theorem**

**1.**

**Proof.**

**Lemma**

**3.**

**Proof.**

**Lemma**

**4.**

**Proof.**

**Corollary**

**1.**

**Proof.**

**Theorem**

**2.**

**Proof.**

**Theorem**

**3.**

**Proof.**

**Theorem**

**4.**

**Proof.**

**Corollary**

**2.**

**Lemma**

**5.**

**Proof.**

**Lemma**

**6.**

**Proof.**

**Lemma**

**7.**

**Proof.**

**Lemma**

**8.**

**Proof.**

**Lemma**

**9.**

**Proof.**

**Theorem**

**5.**

- (i)
- only the graph ${B}_{n}^{\left(1\right)}(0,n-4,0,0)$ has the minimum value of $\tilde{Lz}$ for $n=5$,
- (ii)
- only the graph ${B}_{n;1}$ has the minimum value of $\tilde{Lz}$ for each $n\in \{6,7\}$,
- (iii)
- only the graph ${B}_{n;2}$ has the minimum value of $\tilde{Lz}$ for each $n\in \{8,9\}$,
- (iv)
- only the graphs ${B}_{n;0},{B}_{n;1},{B}_{n;2},{B}_{n;3}$ have the minimum value of $\tilde{Lz}$ for $n=10$.

**Proof.**

**Theorem**

**6.**

**Proof.**

**Lemma**

**10.**

**Proof.**

**Theorem**

**7.**

**Lemma**

**11.**

**Proof.**

**Theorem**

**8.**

**Lemma**

**12.**

**Proof.**

**Theorem**

**9.**

- (i).
- If $n=2(\xi -1)$, then only (the) 3-regular graph(s) possess(es) the highest value of $\tilde{Lz}$ in ${\mathbb{G}}_{n,\xi}$.
- (ii).
- If $n>2(\xi -1)$, then only the graphs with $(\Delta ,\delta )=(3,2)$ possess the highest value of $\tilde{Lz}$ in ${\mathbb{G}}_{n,\xi}$.

**Proof.**

- (i)
- (ii)

**Theorem**

**10.**

**Proof.**

**Theorem**

**11.**

**Proof.**

## 4. Extremal Results Concerning Molecular $\mathit{\xi}$-Cyclic Graphs

**Lemma**

**13**

**.**Consider a molecular $(n,m)$-graph G, where $n\ge 5$. Take

- •$\{1,2,4\}$ and G admits exactly one vertex of degree 2 whenever $2m-n\equiv 1\phantom{\rule{4.44443pt}{0ex}}(mod\phantom{\rule{0.277778em}{0ex}}3)$;
- •$\{1,3,4\}$ and G admits exactly one vertex of degree 3 whenever $2m-n\equiv 2\phantom{\rule{4.44443pt}{0ex}}(mod\phantom{\rule{0.277778em}{0ex}}3)$;
- •$\{1,4\}$ whenever $2m-n\equiv 0\phantom{\rule{4.44443pt}{0ex}}(mod\phantom{\rule{0.277778em}{0ex}}3)$.

**Theorem**

**12.**

**Proof.**

**Corollary**

**3.**

- (i)
- $\{1,2,4\}$ and admitting exactly one vertex of degree 2 possess the least value of $\tilde{Lz}$, when $n\equiv 0\phantom{\rule{4.44443pt}{0ex}}(mod\phantom{\rule{0.277778em}{0ex}}3)$;
- (ii)
- $\{1,3,4\}$ and admitting exactly one vertex of degree 3 possess the least value of $\tilde{Lz}$, when $n\equiv 1\phantom{\rule{4.44443pt}{0ex}}(mod\phantom{\rule{0.277778em}{0ex}}3)$;
- (iii)
- $\{1,4\}$ possess the least value of $\tilde{Lz}$, when $n\equiv 2\phantom{\rule{4.44443pt}{0ex}}(mod\phantom{\rule{0.277778em}{0ex}}3)$.

## 5. Extremal Results for $\mathit{n}$-Order Graphs

**Lemma**

**14**

**.**If G is an n-order graph, then

**Proposition**

**3.**

**Proof.**

**Theorem**

**13.**

**Proof.**

**Lemma**

**15.**

**Proof.**

**Theorem**

**14.**

- (i)
- $\{1,2,4\}$ and admitting exactly one vertex of degree 2 possess the least value of $\tilde{Lz}$, when $n\equiv 0\phantom{\rule{4.44443pt}{0ex}}(mod\phantom{\rule{0.277778em}{0ex}}3)$;
- (ii)
- $\{1,3,4\}$ and admitting exactly one vertex of degree 3 possess the least value of $\tilde{Lz}$, when $n\equiv 1\phantom{\rule{4.44443pt}{0ex}}(mod\phantom{\rule{0.277778em}{0ex}}3)$;
- (iii)
- $\{1,4\}$ possess the least value of $\tilde{Lz}$, when $n\equiv 2\phantom{\rule{4.44443pt}{0ex}}(mod\phantom{\rule{0.277778em}{0ex}}3)$.

**Proof.**

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 2.**The graphs possessing the lowest value of $\tilde{Lz}$ in the set of all n-order connected unicyclic graphs, for $n=5,6,7$.

**Figure 3.**The n-order bicyclic graph ${B}_{n}^{\left(1\right)}({\eta}_{1},{\eta}_{2},{\eta}_{3},{\eta}_{4})$, where ${\eta}_{1}+{\eta}_{2}+{\eta}_{3}+{\eta}_{4}=n-4\ge 1$ with ${\eta}_{1}\ge {\eta}_{4}\ge 0$ and ${\eta}_{2}\ge {\eta}_{3}\ge 0$.

**Figure 4.**The n-order bicyclic graph ${B}_{n}^{\left(2\right)}({\eta}_{1},{\eta}_{2},{\eta}_{3},{\eta}_{4},{\eta}_{5})$, where ${\eta}_{1}+{\eta}_{2}+{\eta}_{3}+{\eta}_{4}+{\eta}_{5}=n-5\ge 0$ with ${\eta}_{1}\ge max\{{\eta}_{2},{\eta}_{3},{\eta}_{4}\}$ and ${\eta}_{i}\ge 0$ for every $i\in \{1,2,\dots ,5\}$.

**Figure 5.**The n-order bicyclic graphs ${B}_{n;\alpha}$ (on the left side) and ${B}_{n}^{*}$ (on the right side), where $0\le \alpha \le \lfloor (n-4)/2\rfloor $.

**Figure 8.**The graphs ${J}_{1},{J}_{2},\dots ,{J}_{5},$ (from left to right, respectively) used in Theorem 6.

**Figure 9.**All connected bicyclic graphs of order 5 and minimum degree at least 2, together with the values of $\tilde{Lz}$.

**Figure 10.**All connected tricyclic graphs of order 5 and minimum degree at least 2. The first, second, and third graphs (from left to right) have the following values of $\tilde{Lz}$, respectively: 20, 22, and 24.

**Figure 11.**The n-order connected tricyclic graphs ${G}^{\u2605}$ of maximum degree $n-1$ and minimum degree no less than 2, for $n=6,7$.

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**MDPI and ACS Style**

Ali, A.; Shang, Y.; Dimitrov, D.; Réti, T.
Ad-Hoc Lanzhou Index. *Mathematics* **2023**, *11*, 4256.
https://doi.org/10.3390/math11204256

**AMA Style**

Ali A, Shang Y, Dimitrov D, Réti T.
Ad-Hoc Lanzhou Index. *Mathematics*. 2023; 11(20):4256.
https://doi.org/10.3390/math11204256

**Chicago/Turabian Style**

Ali, Akbar, Yilun Shang, Darko Dimitrov, and Tamás Réti.
2023. "Ad-Hoc Lanzhou Index" *Mathematics* 11, no. 20: 4256.
https://doi.org/10.3390/math11204256