This is a survey of distance-regular graphs. We present an introduction to distance-regular graphs for the reader who is unfamiliar with the subject, and then give an overview of some developments in the area of distance-regular graphs since the monograph 'BCN' [Brouwer, A.E., Cohen, A.M., Neumaier, A.,  Distance-Regular Graphs , Springer-Verlag, Berlin, 1989] was written.

Distance-regular graphs

Despite more than two decades of publications that offer more innovative model-based designs, the classical 3+3 design remains the most dominant phase I trial design in practice. In this article, we introduce a new trial design, the Bayesian optimal interval (BOIN) design. The BOIN design is easy to implement in a way similar to the 3+3 design, but is more flexible for choosing the target toxicity rate and cohort size and yields a substantially better performance that is comparable to that of more complex model-based designs. The BOIN design contains the 3+3 design and the accelerated titration design as special cases, thus linking it to established phase I approaches. A numerical study shows that the BOIN design generally outperforms the 3+3 design and the modified toxicity probability interval (mTPI) design. The BOIN design is more likely than the 3+3 design to correctly select the maximum tolerated dose (MTD) and allocate more patients to the MTD. Compared to the mTPI design, the BOIN design has a substantially lower risk of overdosing patients and generally a higher probability of correctly selecting the MTD. User-friendly software is freely available to facilitate the application of the BOIN design.

/pdf/bayesian-optimal-interval-design-a-simple-and-well-fnf2m434hf.pdf

Bayesian Optimal Interval Design: A Simple and Well-Performing Design for Phase I Oncology Trials

Purpose: MLN8237, a selective small-molecule inhibitor of Aurora kinase A, has activity in a broad range of preclinical pediatric cancer models. We conducted a phase I trial in children with refractory/recurrent solid tumors to define the maximum-tolerated dose, toxicities, and pharmacokinetic properties of MLN8237. Experimental Design: MLN8237 was administered orally either once daily or divided twice daily for seven days, every 21 days. Using a rolling-six design, four dose levels (45, 60, 80, and 100 mg/m 2 /day) were evaluated on the once-daily schedule, and two dose levels (60 and 80 mg/m 2 /d) on the twice-daily schedule. Pharmacokinetic studies were conducted with the initial dose and trough drug concentrations also measured at the steady state. Results: Thirty-seven patients were enrolled. On the once-daily dosing schedule, myelosuppression was dose limiting in three of four patients at 100 mg/m 2 , and one of six patients had dose-limiting mood alteration at 80 mg/m 2 . At 45 mg/m 2 , one of six patients experienced dose-limiting mucositis. Mucositis and myelosuppression were dose limiting at 80 mg/m 2 on the twice-daily schedule, and one of five patients at 60 mg/m 2 on the twice-daily schedule experienced a dose-limiting alkaline phosphatase. Five of 11 patients experienced hand–foot–skin syndrome with twice-daily dosing versus one of 21 after once-daily dosing. There was one partial response and six with prolonged stable disease among 33 evaluable subjects. Conclusion: The twice-daily dose regimen is well tolerated in adults; however, children experienced a greater frequency of myelosuppression and hand–foot–skin syndrome on this schedule. Children tolerated a higher dose and the recommended pediatric phase II dose is 80 mg/m 2 /d once daily for seven days. Clin Cancer Res; 18(21); 6058–64. ©2012 AACR .

https://clincancerres.aacrjournals.org/content/clincanres/early/2012/09/15/1078-0432.CCR-11-3251.full.pdf

Pediatric Phase I Trial and Pharmacokinetic Study of MLN8237, an Investigational Oral Selective Small-Molecule Inhibitor of Aurora Kinase A: A Children's Oncology Group Phase I Consortium Study

Late-onset toxicity is common for novel molecularly targeted agents and immunotherapy. It causes major logistic difficulty for existing adaptive phase I trial designs, which require the observance of toxicity early enough to apply dose-escalation rules for new patients. The same logistic difficulty arises when the accrual is rapid. We propose the time-to-event Bayesian optimal interval (TITE-BOIN) design to accelerate phase I trials by allowing for real-time dose assignment decisions for new patients while some enrolled patients' toxicity data are still pending. Similar to the rolling six design, the TITE-BOIN dose-escalation/deescalation rule can be tabulated before the trial begins, making it transparent and simple to implement, but is more flexible in choosing the target dose-limiting toxicity (DLT) rate and has higher accuracy to identify the MTD. Compared with the more complicated model-based time-to-event continuous reassessment method (TITE-CRM), the TITE-BOIN has comparable accuracy to identify the MTD but is simpler to implement with substantially better overdose control. As the TITE-CRM is more aggressive in dose escalation, it is less likely to underdose patients. When there are no pending data, the TITE-BOIN seamlessly reduces to the BOIN design. Numerical studies show that the TITE-BOIN design supports continuous accrual without sacrificing patient safety or the accuracy of identifying the MTD, and therefore has great potential to accelerate early-phase drug development. Clin Cancer Res; 24(20); 4921-30. ©2018 AACR.

https://clincancerres.aacrjournals.org/content/clincanres/24/20/4921.full.pdf

Time-to-Event Bayesian Optimal Interval Design to Accelerate Phase I Trials.

On the strong metric dimension of corona product graphs and join graphs

A resolving set of a graph is a set of vertices with the property that the list of distances from any vertex to those in the set uniquely identifies that vertex. In this paper, we construct a resolving set of Johnson graphs, doubled Odd graphs, doubled Grassmann graphs and twisted Grassmann graphs, respectively, and obtain the upper bounds on the metric dimension of these graphs.

Metric dimension of some distance-regular graphs

Association schemes based on maximal isotropic subspaces in singular classical spaces

Assume that @n is a positive integer and @d=0,1 or 2. In this paper we introduce the orthogonal graph @C^2^@n^+^@d over a Galois ring of odd characteristic and prove that it is arc transitive. Moreover, we compute its parameters as a quasi-strongly regular graph. In particular, we show that @C^2^+^@d is a strongly regular graph and @C^2^@n^+^1 is a strictly Deza graph when @n>=2.

Orthogonal graphs over Galois rings of odd characteristic

Assume that $\nu$ is a positive integer and $\delta=0, 1$ or $2$. In this paper we introduce the orthogonal graph $\Gamma^{2\nu+\delta}$ over a Galois ring of odd characteristic and prove that it is arc transitive. Moreover, we compute its parameters as a quasi-strongly regular graph. In particular, we show that $\Gamma^{2+\delta}$ is a strongly regular graph and $\Gamma^{2\nu+1}$ is a strictly Deza graph when $\nu\geq 2$.

Suborbits of a point stabilizer in the orthogonal group on the last subconstituent of orthogonal dual polar graphs

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Papers

Metric dimension of some distance-regular graphs

Association schemes based on maximal isotropic subspaces in singular classical spaces

Orthogonal graphs over Galois rings of odd characteristic

Orthogonal graphs over Galois rings of odd characteristic

Suborbits of a point stabilizer in the orthogonal group on the last subconstituent of orthogonal dual polar graphs