# The Differential on Graph Operator Q(G)

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## Abstract

**:**

## 1. Introduction

“You are allowed to buy as many tokens as you like, say k tokens, at cost of $1 each. You then place the tokens on some subset of k vertices of G. For each vertex of G which has no token on it, but is adjacent to a vertex with a token on it, you receive $1. Your objective is to maximize your profit, that is, the total value received minus the cost of the tokens bought.“

- DIF: If G is a graph and k is an integer, decide if $\partial \left(G\right)\ge k$;
- ROM: If G is a graph and l is an integer, decide if ${\gamma}_{R}\left(G\right)\le l$.

- To obtain inequalities relating the differential of G with other graph invariants of G.
- To study the differential $\partial \left(G\right)$ for the most important graph families (complete graphs, trees, star graphs, etc.).
- To study the variation of the differential $\partial \left(G\right)$ under certain transformations (graph operators, products of graphs, etc.).
- To determine the computational complexity of the DIF problem.

**Remark**

**1.**

## 2. Results

**Proposition**

**1.**

- (i)
- $\left|V\right(\mathrm{Q}\left(G\right)\left)\right|=\left|V\right|+\left|L\right|.$
- (ii)
- $\left|E\right(\mathrm{Q}\left(G\right)\left)\right|=2\left|L\right|+\left|E\right(L\left(G\right)\left)\right|.$
- (iii)
- $L\left(G\right)$ is an induced subgraph of $\mathrm{Q}\left(G\right).$
- (iv)
- $G\cong \mathrm{Q}\left(G\right)$ if and only if $G\cong {E}_{n}$.
- (v)
- If $v\in V$, then ${\delta}_{\mathrm{Q}\left(G\right)}\left(v\right)={\delta}_{G}\left(v\right)$.
- (vi)
- If $u\in L$, then $\left|N\right(u)\cap V|=2$.
- (vii)
- If $v\in V$ and $u\in {N}_{\mathrm{Q}\left(G\right)}\left(v\right)$, then ${N}_{\mathrm{Q}\left(G\right)}\left[v\right]\subseteq {N}_{\mathrm{Q}\left(G\right)}\left[u\right]$.
- (viii)
- If $u\in L$, then there exist vertices $x,y\in V$ such that $\{x,y\}={N}_{\mathrm{Q}\left(G\right)}\left(u\right)\cap V$ and ${\delta}_{\mathrm{Q}\left(G\right)}\left(u\right)={\delta}_{G}\left(x\right)+{\delta}_{G}\left(y\right)$.
- (ix)
- If G is a regular graph, then $\mathrm{Q}\left(G\right)$ is a biregular graph.

**Proposition**

**2.**

**Proof.**

**Proposition**

**3.**

**Proof.**

**Proposition**

**4.**

**Proof.**

**Proposition**

**5.**

**Proof.**

**Corollary**

**1.**

**Corollary**

**2.**

**Proposition**

**6.**

**Proof.**

**Observation**

**1.**

**Proposition**

**7.**

**Proof.**

**Proposition**

**8**

**Proposition**

**9**

- (a)
- $\mathrm{\Delta}=n-1$ iff $\partial \left(G\right)=n-2$.
- (b)
- $\mathrm{\Delta}=n-2$ iff $\partial \left(G\right)=n-3$.
- (c)
- If $\mathrm{\Delta}=n-3$, then $\partial \left(G\right)=n-4$.

**Proposition**

**10.**

- (a)
- $\partial \left(\mathrm{Q}\left(G\right)\right)=\left|V\right(\mathrm{Q}\left(G\right)\left)\right|-2$ iff $G={P}_{2}$.
- (b)
- $\partial \left(\mathrm{Q}\left(G\right)\right)=\left|V\right(\mathrm{Q}\left(G\right)\left)\right|-3$ iff $G\in \{{P}_{3},{K}_{3},{E}_{1}\uplus {P}_{2}\}$.

**Proof.**

**Theorem**

**1.**

**Proof.**

**Observation**

**2.**

**Corollary**

**3.**

**Corollary**

**4.**

**Observation**

**3.**

**Proposition**

**11.**

**Proof.**

**Corollary**

**5.**

**Theorem**

**2.**

**Proof.**

**Corollary**

**6.**

- (i)
- $\partial \left(\mathrm{Q}\left({K}_{n}\right)\right)={\displaystyle \frac{n(n-1)}{2}}$.
- (ii)
- $\partial \left(\mathrm{Q}\left({S}_{n}\right)\right)=n-1$.
- (iii)
- $\partial \left(\mathrm{Q}\left({S}_{m,n}\right)\right)=n+m-1$.
- (iv)
- $\partial \left(\mathrm{Q}\left({K}_{m,n}\right)\right)=mn$.
- (v)
- $\partial \left(\mathrm{Q}\left({P}_{n}\right)\right)=n-1$.
- (vi)
- $\partial \left(\mathrm{Q}\left({C}_{n}\right)\right)=n$.
- (vii)
- $\partial \left(\mathrm{Q}\left({W}_{n}\right)\right)=2(n-1)$.
- (viii)
- $\partial \left(\mathrm{Q}\left(T\right)\right)=n-1$.

**Proof.**

## 3. Concluding Remarks

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 2.**Here, Gis the star graph ${S}_{5}$. Note that $\partial \left(\mathrm{Q}\left({S}_{5}\right)\right)=4$ and $\partial \left(L\left({S}_{5}\right)\right)$ = 2.

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**MDPI and ACS Style**

Basilio, L.A.; Simon, J.C.; Leaños, J.; Cayetano, O.R.
The Differential on Graph Operator Q(G). *Symmetry* **2020**, *12*, 751.
https://doi.org/10.3390/sym12050751

**AMA Style**

Basilio LA, Simon JC, Leaños J, Cayetano OR.
The Differential on Graph Operator Q(G). *Symmetry*. 2020; 12(5):751.
https://doi.org/10.3390/sym12050751

**Chicago/Turabian Style**

Basilio, Ludwin A., Jair Castro Simon, Jesús Leaños, and Omar Rosario Cayetano.
2020. "The Differential on Graph Operator Q(G)" *Symmetry* 12, no. 5: 751.
https://doi.org/10.3390/sym12050751