No abstract

In this article, we give sharp bounds on the Hosoya index and the Merrifield-Simmons index for connected graphs of fixed size. As a consequence, we determine all connected graphs of any fixed order and size which maximize the Merrifield-Simmons index. Sharp lower bounds on the Hosoya index are known for graphs of order n and size m ∈ [n − 1, 2n − 3] ∪ n−1 2 , n 2 ; while sharp upper bounds were only known for graphs of order n and size m ≤ n + 2. We give sharp upper bounds on the Hosoya index for dense graphs with m ≥ n 2 − 2n/3. Moreover, all extreme graphs are also determined.

Chinese Remainder Theorem (CRT) has been widely studied with its applications in frequency estimation, phase unwrapping, coding theory and distributed data storage. Since traditional CRT is greatly sensitive to the errors in residues due to noises, the problem of robustly reconstructing integers via the erroneous residues has been intensively studied in the literature. In order to robustly reconstruct integers, there are two kinds of traditional methods: the one is to introduce common divisors in the moduli and the other is to directly decrease the dynamic range. In this paper, we take further insight into the geometry property of the linear space associated with CRT. Echoing both ways to introduce redundancy, we propose a pseudo metric to analyze the trade-off between the error bound and the dynamic range for robust CRT in general. Furthermore, we present the first robust CRT for multiple numbers to solve the problem of the CRT-based undersampling frequency estimation in general cases. Based on symmetric polynomials, we proved that in most cases, the problem can be solved in polynomial time efficiently. The work in this paper is towards a complete theoretical solution to the open problem over 20 years.