Abstract
DoΕ‘liΔ et al. defined the Mostar index of a graph πΊ as ππ(πΊ) =
Ξ£
π’π£βπΈ(πΊ)
|ππΊ(π’, π£) β ππΊ(π£, π’)|, where,
for an edge π’π£ of πΊ, the term ππΊ(π’, π£) denotes the number of vertices of πΊ that have a smaller
distance in πΊ to π’ than to π£. For a graph πΊ of order π and maximum degree at most π₯, we
show ππ(πΊ) β€ π₯2 π2 β (1 β π(1))ππ₯π log(log(π)), where ππ₯ > 0 only depends on π₯ and the π(1) term
only depends on π. Furthermore, for integers π0 and π₯ at least 3, we show the existence of a
π₯-regular graph of order π at least π0 with ππ(πΊ) β₯ π₯2 π2 βπβ²
π₯
π log(π), where πβ²
π₯
> 0 only depends
on π₯.