Combinatorics (math.CO)

Abstract: Binomial coefficients and harmonic numbers are important in many branches of number theory. With the help of the operator method and several summation and transformation formulas for hypergeometric series, we prove eight conjectural series of Z.-W. Sun containing binomial coefficients and harmonic numbers in this paper.

Abstract: A subset $S$ of a group $G$ is $t$-weakly sequenceable if there is an ordering $(a_1, \ldots, a_k)$ of its elements such that the partial sums $(s_0, s_1, \ldots, s_k)$, given by $s_0 = 0$ and $s_i = \sum_{j=1}^i a_j$ for $1 \leq i \leq k$, are different whenever $i$ and $j$ are distinct and $|i-j|\leq t$. In [10] it was proved that if the order of a group is $pe$ then all sufficiently large subsets of the non-identity elements are $t$-weakly sequenceable when $p > 3$ is prime, $e \leq 3$ and $t \leq 6$. Inspired by this result, we show that, if $G$ is of the type $G = \mathbb{Z}_p \rtimes_{\varphi} \mathbb{Z}_2$ and the set $S$ is balanced (i.e. contains the same number of even elements which are those in the coset $\mathbb{Z}_p\rtimes_{\varphi} \{0\}$ and odd ones that are those in its complement) then $S$ admits, regardless of its size, an alternating parity $t$-weak sequencing whenever $p > 3$ is prime and $t \leq 8$. On the other hand, we have been able to prove also the following asymptotic result. Let us consider groups of type $G = H \rtimes_{\varphi} \mathbb{Z}_2$, then all sufficiently large balanced subsets of the non-identity elements admit an alternating parity $t$-weak sequencing. This result has been obtained using a hybrid approach that combines both Ramsey theory and the probabilistic method. The same procedure works also for studying the weak sequenceability for generic sufficiently large (not necessarily balanced) sets. Here we have been able to prove that, if the size of a subset $S$ of a group $G$ is large enough and if $S$ does not contain $0$, then $S$ is $t$-weakly sequenceable.

Abstract: Belk and Connelly introduced the realizable dimension $\textrm{rd}(G)$ of a finite graph $G$, which is the minimum nonnegative integer $d$ such that every framework $(G,p)$ in any dimension admits a framework in $\mathbb{R}^d$ with the same edge lengths. They characterized finite graphs with realizable dimension at most $1$, $2$, or $3$ in terms of forbidden minors. In this paper, we consider periodic frameworks and extend the notion to $\mathbb{Z}$-symmetric graphs. We give a forbidden minor characterization of $\mathbb{Z}$-symmetric graphs with realizable dimension at most $1$ or $2$, and show that the characterization can be checked in polynomial time for given quotient $\mathbb{Z}$-labelled graphs.

Abstract: For a given connected graph $G$, the edge Mostar index $Mo_e(G)$ is defined as $Mo_e(G)=\sum_{e=uv \in E(G)}|m_u(e|G) - m_v(e|G)|$, where $m_u(e|G)$ and $m_v(e|G)$ are respectively, the number of edges of $G$ lying closer to vertex $u$ than to vertex $v$ and the number of edges of $G$ lying closer to vertex $v$ than to vertex $u$. In this paper, we determine sharp upper bound for edge Mostar index on bicyclic graphs with fixed number of edges, also the graphs that achieve the bound are completely characterized, and thus disprove a conjecture on the edge Mostar index of bicyclic graph in [H. Liu, L. Song, Q. Xiao, Z. Tang, On edge Mostar index of graphs. Iranian J. Math. Chem. 11(2) (2020) 95--106].