Some inequalities in distance-regular graphs
For integers \(r\) \((>0)\) and \(a\), let \(n_ r(a)\) denote the following value \[ n_ r(a)= 1+ a+ a(a- 1)+\cdots+ a(a-1)^ i+\cdots+ a(a-1)^ r. \] In this paper, we prove the following theorem and give several corollaries. Theorem. Let \(G\) be a distance-regular graph with girth \(g\) and diameter \(d\). Let \(a_ i\), \(b_ i\), \(c_ i\) denote the usual intersection numbers. Let \(r\), \(s\), \(t\) be non-negative integers with \(2r+ 3\leq g\), \(r<s\), \(s+ t\leq d\). Then \[ n_ r(a_ s)\leq {b_ t b_{t+1}\cdots b_{t+s-1}\over c_{r+1} c_{r+2}\cdots c_ s},\quad\text{if }c_ s= c_{s+t},\tag{i} \] \[ n_ r(a_{s+t})\leq {b_ t b_{t+1}\cdots b_{t+s-1}\over c_{r+1} c_{r+2}\cdots c_ s},\quad\text{if }b_ s= b_{s+t},\tag{ii} \] \[ n_ r(a_ t)\leq {c_{t+1}c_{t+2}\cdots c_{t+s}\over c_{r+1}c_{r+2}\cdots c_ s},\quad\text{if }b_ s= c_ t.\tag{iii} \] {}.
scientific article; zbMATH DE number 572447