arXiv subjects

Sumin Huang

Publications and source records attributed to Sumin Huang.

6 recordsLinked to original sources

arXiv2024

The maximum sum of the sizes of all intersections within $m$-size families

For a family of sets $\mathcal{F}$, let $\omega(\mathcal{F}):=\sum_{\{A,B\}\subset \mathcal{F}}|A\cap B|$. In this paper, we prove that provided $n$ is sufficiently large, for any $\mathcal{F}\subset \binom{[n]}{k}$ with $|\mathcal{F}|=m$, $\omega(\mathcal{F})$ is maximized by the family consisting of the first $m$ sets in the lexicographical ordering on $\binom{[n]}{k}$. Compared to the maximum number of adjacent pairs in families, determined by Das, Gan and Sudakov in 2016, $\omega(\mathcal{F})$ distinguishes the contributions of intersections of different sizes. Then our results is an extension of Ahlswede and Katona's results in 1978, which determine the maximum number of adjacent edges in graphs. Besides, since $\omega(\mathcal{F})=\frac{1}{2}\left(\sum_{x\in [n]}|\{F\in \mathcal{F}:x\in F\}|^2-km\right)$ for $k$-uniform family with size $m$, our results also give a sharp upper bound of the sum of squares of degrees in a hypergraph.

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arXiv2024

Pairwise Intersection Profiles of Intersecting Families

Let $\mathcal F\subseteq\binom{[n]}{k}$ be a $k$-uniform family, and let $P_m(\mathcal F)$ denote the number of ordered pairs $(A,B)\in\mathcal F\times\mathcal F$ with $|A\cap B|=m$. We study the pairwise intersection profile $(P_0(\mathcal F),P_1(\mathcal F),\ldots,P_k(\mathcal F))$ of intersecting families. We prove that for every $1\leq m\leq k$, if $n\geq\max\{2k,3k-m-1\}$, then $$ P_m(\mathcal F)\leq P_m(\mathcal S_1^k), $$ where $\mathcal S_1^k$ is a full star. Moreover, if $n\geq \max\{2k+1,3k-m\}$, then equality holds if and only if $\mathcal F$ is a star up to permutations. As a corollary, for any function $f:\{0,1,\ldots,k\}\rightarrow \mathbb{R}_{\geq 0}$ and $n\geq 3k-2$, we show that $\Phi_f(\mathcal F)=\sum_{(A,B)\in\mathcal F\times\mathcal F}f(|A\cap B|)$ is maximized by a star. We also establish a stability result showing that, for $n\geq \max\{2k+1,3k-m\}$, if $P_m(\mathcal F)$ is sufficiently close to its maximum, then $\mathcal F$ has an element of degree close to $\binom{n-1}{k-1}$.

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arXiv2023

An embedding technique in the study of word-representabiliy of graphs

Word-representable graphs, which are the same as semi-transitively orientable graphs, generalize several fundamental classes of graphs. In this paper we propose a novel approach to study word-representability of graphs using a technique of homomorphisms. As a proof of concept, we apply our method to show word-representability of the simplified graph of overlapping permutations that we introduce in this paper. For another application, we obtain results on word-representability of certain subgraphs of simplified de Bruijn graphs that were introduced recently by Petyuk and studied in the context of word-representability.

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arXiv2022

The list-coloring function of signed graphs

It is known that, for any $k$-list assignment $L$ of a graph $G$, the number of $L$-list colorings of $G$ is at least the number of the proper $k$-colorings of $G$ when $k>(m-1)/\ln(1+\sqrt{2})$. In this paper, we extend the Whitney's broken cycle theorem to $L$-colorings of signed graphs, by which we show that if $k> \binom{m}{3}+\binom{m}{4}+m-1$ then, for any $k$-assignment $L$, the number of $L$-colorings of a signed graph $\Sigma$ with $m$ edges is at least the number of the proper $k$-colorings of $\Sigma$. Further, if $L$ is $0$-free (resp., $0$-included) and $k$ is even (resp., odd), then the lower bound $\binom{m}{3}+\binom{m}{4}+m-1$ for $k$ can be improved to $(m-1)/\ln(1+\sqrt{2})$.

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arXiv2021

The maximum number of stars in a graph without linear forest

For two graphs $J$ and $H$, the generalized Tur\'{a}n number, denoted by $ex(n,J,H)$, is the maximum number of copies of $J$ in an $H$-free graph of order $n$. A linear forest $F$ is the disjoint union of paths. In this paper, we determine the number $ex(n,S_r,F)$ when $n$ is large enough and characterize the extremal graphs attaining $ex(n,S_r,F)$, which generalizes the results on $ex(n, S_r, P_k)$, $ex(n,K_2,(k+1) P_2)$ and $ex(n,K^*_{1,r},(k+1) P_2)$. Finally, we pose the problem whether the extremal graph for $ex(n,J,F)$ is isomorphic to that for $ex(n,S_r,F)$, where $J$ is any graph such that the number of $J$'s in any graph $G$ does not decrease by shifting operation on $G$.

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arXiv2019

On the resistance distance and Kirchhoff index of a linear hexagonal (cylinder) chain

The resistance between two nodes in some resistor networks has been studied extensively by mathematicians and physicists. Let $L_n$ be a linear hexagonal chain with $n$\, 6-cycles. Then identifying the opposite lateral edges of $L_n$ in ordered way yields the linear hexagonal cylinder chain, written as $R_n$. We obtain explicit formulae for the resistance distance $r_{L_n}(i, j)$ (resp. $r_{R_n}(i,j)$) between any two vertices $i$ and $j$ of $L_n$ (resp. $R_n$). To the best of our knowledge $\{L_n\}_{n=1}^{\infty}$ and $\{R_n\}_{n=1}^{\infty}$ are two nontrivial families with diameter going to $\infty$ for which all resistance distances have been explicitly calculated. We determine the maximum and the minimum resistance distances in $L_n$ (resp. $R_n$). The monotonicity and some asymptotic properties of resistance distances in $L_n$ and $R_n$ are given. As well we give formulae for the Kirchhoff indices of $L_n$ and $R_n$ respectively.

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