The $(\alpha ,\beta ,s,t)$-diameter of graphs: A particular case of conditional diameter

Abstract

The conditional diameter of a connected graph $\Gamma =(V,E)$ is defined as follows: given a property $\mathcal{P}$ of a pair $({\Gamma }_1,{\Gamma }_2)$ of subgraphs of $\Gamma$, the so-called conditional diameter or $\mathcal{P}$-diameter measures the maximum distance among subgraphs satisfying $\mathcal{P}$. That is,

$$D_{\mathcal{P}}(\Gamma )≔\underset{{\Gamma }_1,{\Gamma }_2\subset \Gamma }{\max }\{\mathrm{\partial }({\Gamma }_1,{\Gamma }_2):{\Gamma }_1,{\Gamma }_2\mathrm{satisfy}\mathcal{P}\}\text{.}$$

In this paper we consider the conditional diameter in which $\mathcal{P}$ requires that $\delta (u)⩾\alpha$ for all $u\in V({\Gamma }_1)$, $\delta (v)⩾\beta$ for all $v\in V({\Gamma }_2)$, $|V({\Gamma }_1)|⩾s$ and $|V({\Gamma }_2)|⩾t$ for some integers $1⩽s,t⩽\left|V\right|$ and $\delta ⩽\alpha ,\beta ⩽\Delta$, where $\delta (x)$ denotes the degree of a vertex x of $\Gamma$, $\delta$ denotes the minimum degree and $\Delta$ the maximum degree of $\Gamma$. The conditional diameter obtained is called $(\alpha ,\beta ,s,t)$-diameter. We obtain upper bounds on the $(\alpha ,\beta ,s,t)$-diameter by using the k-alternating polynomials on the mesh of eigenvalues of an associated weighted graph. The method provides also bounds for other parameters such as vertex separators.