The (a,b,s,t)-diameter of graphs: a particular case of conditional diameter

The conditional diameter of a connected graph $\mathrm{\Gamma }=(V,E)$ is defined as follows: given a property $\mathcal{P}$ of a pair $({\mathrm{\Gamma }}_1,{\mathrm{\Gamma }}_2)$ of subgraphs of $\mathrm{\Gamma }$, the so-called \emph{conditional diameter} or ${\cal P}$-\text{em}diametermeasuresthemaximumdistanceamongsubgraphssatisfying${\cal P}$. That is, $$D_{\mathcal{P}}(\mathrm{\Gamma }):=\underset{{\mathrm{\Gamma }}_1,{\mathrm{\Gamma }}_2\subset \mathrm{\Gamma }}{\max }\{\mathrm{\partial }({\mathrm{\Gamma }}_1,{\mathrm{\Gamma }}_2):{\mathrm{\Gamma }}_1,{\mathrm{\Gamma }}_2\mathrm{s}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{s}\mathrm{f}\mathrm{y}\mathcal{P}\}\mathrm{.}$$ In this paper we consider the conditional diameter in which $\mathcal{P}$ requires that $\delta (u)\ge \alpha$ for all $ u\in V(\Gamma_1)$,$\delta(v)\ge \beta$forall$v\in V(\Gamma_2)$,$| V(\Gamma_1)| \ge s$and$| V(\Gamma_2)| \ge t$forsomeintegers$1\le s,t\le |V|$and$\delta \le \alpha, \beta \le \Delta$,where$\delta(x)$ denotes the degree of a vertex $x$ of $\mathrm{\Gamma }$, $\delta$ denotes the minimum degree and $\mathrm{\Delta }$ the maximum degree of $\mathrm{\Gamma }$. The conditional diameter obtained is called $(\alpha ,\beta ,s,t)$-\emph{diameter}. We obtain upper bounds on the $(\alpha ,\beta, s,t)$-diameterbyusingthe$k$-alternating polynomials on the mesh of eigenvalues of an associated weighted graph. The method provides also bounds for other parameters such as vertex separators.