# On the Inverse Degree Polynomial

^{1}

^{2}

^{3}

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## Abstract

**:**

## 1. Introduction

## 2. Some Preliminaries

**Proposition**

**1.**

- $I{D}^{\left(j\right)}(G,x)\u2a7e0$ for every $j\u2a7e0$ and $x\in [0,\infty )$,
- $ID(G,x)>0$ on $(0,\infty )$,
- $ID(G,x)$ is strictly increasing on $[0,\infty )$ if and only if G is not isomorphic to an union of path graphs ${P}_{2}$,
- $ID(G,x)$ is strictly convex on $[0,\infty )$ if and only if G is not isomorphic to an union of path graphs and/or cycle graphs, and
- $k=ID(G,0)\u2a7dID(G,x)\u2a7dID(G,1)=n$ for every $x\in [0,1]$.

**Proposition**

**2.**

**Proposition**

**3.**

## 3. Main Results

**Proposition**

**4.**

- $ID(G,1)=n$,
- $I{D}^{\prime}(G,1)=2m-n$,
- $I{D}^{\u2033}(G,1)={M}_{1}\left(G\right)-6m+2n$, and
- $I{D}^{\u2034}(G,1)=F\left(G\right)-6{M}_{1}\left(G\right)+22m-6n$.

**Proof.**

**Corollary**

**1.**

- $n=ID(G,1)$,
- $m=\frac{1}{2}\left(I{D}^{\prime}(G,1)+ID(G,1)\right)$,
- ${M}_{1}\left(G\right)=I{D}^{\u2033}(G,1)+3I{D}^{\prime}(G,1)+ID(G,1)$, and
- $F\left(G\right)=I{D}^{\u2034}(G,1)+6I{D}^{\u2033}(G,1)+7I{D}^{\prime}(G,1)+ID(G,1)$.

**Corollary**

**2.**

**Proposition**

**5.**

**Proof.**

**Proposition**

**6.**

**Proof.**

**Proposition**

**7.**

**Proof.**

**Proposition**

**8.**

- $DegID(G,x)=\Delta -1$.
- ${Deg}_{min}ID(G,x)=\delta -1$.
- $x=0$ is a zero of $ID(G,x)$ if and only if $\delta >1$.
- If $\delta >1$, then $x=0$ is a zero of $ID(G,x)$ with multiplicity $\delta -1$.
- $DegID(G,x)\u2a7dn-2$, and the equality is attained if and only if there exists at least a dominant vertex in G.
- If Γ is a subgraph of G, then $DegID(\mathsf{\Gamma},x)\u2a7dDegID(G,x)$ and ${Deg}_{min}ID(\mathsf{\Gamma},x)\u2a7d{Deg}_{min}ID(G,x)$.

**Proof.**

**Theorem**

**1.**

**Proof.**

**Theorem**

**2.**

**Proof.**

**Theorem**

**3**

**Theorem**

**4.**

**Theorem**

**5.**

- i.
- If $\delta \ge \frac{(n-i)}{2}$, then $diamG\le i+1,\phantom{\rule{4pt}{0ex}}\forall i=1,\dots ,5$.
- ii.
- If $\delta \ge \frac{(n-2)}{3}$, then $diamG\le 5$.
- iii.
- If $\delta \ge \frac{(n-3)}{3}$, then $diamG\le 6$.

**Proposition**

**9.**

- i.
- If $\delta =1$ we have $diam\mathsf{\Gamma}\le n-1-(r-1)(\delta +1)$.
- ii.
- If $\delta \ge 2$, then$$diam\mathsf{\Gamma}\le max\left\{2,\u230a\frac{3n-4}{\delta +1}\u230b-3r+2\right\}.$$

**Proof.**

**Corollary**

**3.**

**Proposition**

**10.**

- If $\delta \ge (n-r)/(r+1)$, then $diam\mathsf{\Gamma}\le 2.$
- If $\delta \ge (n-r-1)/(r+1)$, then $diam\mathsf{\Gamma}\le 3.$
- If $\delta \ge (n-r-2)/(r+1)$, then $diam\mathsf{\Gamma}\le 4$.
- If $\delta \ge (n-r-3)/(r+1)$, then $diam\mathsf{\Gamma}\le 5$.
- If $\delta \ge (n-r-4)/(r+1)$, then $diam\mathsf{\Gamma}\le 6$.
- If $\delta \ge (n-r-1)/(r+2)$, then $diam\mathsf{\Gamma}\le 5$.
- If $\delta \ge (n-r-2)/(r+2)$, then $diam\mathsf{\Gamma}\le 6$.

**Proof.**

**Proposition**

**11.**

- i.
- If $ID(G,0)\ne 0$ then $diam\mathsf{\Gamma}\le ID(G,1)-1-(r-1)({Deg}_{min}ID(G,x)+2)$,
- ii.
- If $ID(G,0)=0$, then$$diam\mathsf{\Gamma}\le max\left\{2,\u230a\frac{3ID(G,1)-4}{{Deg}_{min}ID(G,x)+2}\u230b-3r+2\right\}.$$

**Proof.**

**Proposition**

**12.**

**Proposition**

**13.**

- i.
- If $Deg}_{min}ID(G,x)+1\ge \frac{\left(ID\right(G,1)-r+1-j)}{(r+1)$, then $diam\mathsf{\Gamma}\le j+1,\phantom{\rule{4pt}{0ex}}\forall j=1,\dots ,5$.
- ii.
- If $Deg}_{min}ID(G,x)+1\ge \frac{\left(ID\right(G,1)-r-1)}{(r+2)$, then $diam\mathsf{\Gamma}\le 5$.
- iii.
- If $Deg}_{min}ID(G,x)+1\ge \frac{\left(ID\right(G,1)-r-2)}{(r+2)$, then $diam\mathsf{\Gamma}\le 6$.

**Theorem**

**6.**

- $1\u2a7d\mathfrak{D}\left(ID\right(G,x\left)\right)\u2a7dn-\delta -(r-1)(\delta +1)$,
- $\mathfrak{D}\left(ID\right(G,x\left)\right)=1$ if and only if G is regular,
- $\mathfrak{D}\left(ID\right(G,x\left)\right)=n-1$ if and only if G is isomorphic to ${\mathsf{\Gamma}}_{n}$.

**Proof.**

**Corollary**

**4.**

**Proposition**

**14.**

**Proposition**

**15.**

**Proof.**

**Theorem**

**7.**

**Proof.**

**Proposition**

**16.**

**Proof.**

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**MDPI and ACS Style**

Bosch, P.; Rodríguez, J.M.; Rosario, O.; Sigarreta, J.M.
On the Inverse Degree Polynomial. *Symmetry* **2019**, *11*, 1490.
https://doi.org/10.3390/sym11121490

**AMA Style**

Bosch P, Rodríguez JM, Rosario O, Sigarreta JM.
On the Inverse Degree Polynomial. *Symmetry*. 2019; 11(12):1490.
https://doi.org/10.3390/sym11121490

**Chicago/Turabian Style**

Bosch, Paul, José Manuel Rodríguez, Omar Rosario, and José María Sigarreta.
2019. "On the Inverse Degree Polynomial" *Symmetry* 11, no. 12: 1490.
https://doi.org/10.3390/sym11121490