Let G be a connected graph. Gao et al. first introduced the invariant of G: ΔM(G)=Mo(G)−irr(G), and raised a problem: how to determine the extremal graphs among all connected graphs of order n with respect to ΔM(G), where Mo(G) and irr(G) stand for the Mostar index and irregularity of G, respectively. In this paper, we characterize the upper bounds of ΔM(G) over all unicyclic graphs and bicyclic graphs with diameter 3, and determine the extremal graphs.
In this paper, we prove that if n≥4 and a≥0 are integers satisfying a<n3, then
\gcd\left(\left\{\binom{n}{k}:a<k<n-a\right\}\right)=\prod_{n=p^{m}+b(n,p),\ 0\le b(n,p)\leq a,} p, where \binom{n}{k}=\frac{n!}{k!(n-k)!}, and the product in the right hand side runs through all primes p such that n=p^{m}+b(n,p), m\in\mathbb{N} and 0\le b(n,p)\leq a.
As an application of our result, we give an answer to a problem in Hong [16].