Mark MacLean, Paul Terwilliger
We consider a bipartite distance-regular graph $G$ with diameter at least 4 and valency at least 3. Fix a vertex of $G$ and let $T$ denote the corresponding subconstituent algebra. We give a detailed description of a certain type of irreducible $T$-module, said to be thin with endpoint 2.
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We consider a bipartite distance-regular graph $G$ with diameter $D$ at least 4 and valency $k$ at least 3. We obtain upper and lower bounds for the local eigenvalues of $G$ in terms of the intersection numbers of $G$ and the eigenvalues of $G$. Fix a vertex of $G$ and let $T$ denote the corresponding subconstituent algebra. We give a detailed description of those thin irreducible $T$-modules that have endpoint 2 and dimension $D-3$. In an earlier paper the first author defined what it means for $G$ to be taut. We obtain three characterizations of the taut condition, each of which involves the…
Let $Γ$ denote a finite, undirected, connected graph, with vertex set $X$. Fix a vertex $x \in X$. Associated with $x$ is a certain subalgebra $T=T(x)$ of ${\rm Mat}_X(\mathbb C)$, called the subconstituent algebra. The algebra $T$ is semisimple. Hora and Obata introduced a certain subalgebra $Q \subseteq T$, called the quantum adjacency algebra. The algebra $Q$ is semisimple. In this paper we investigate how $Q$ and $T$ are related. In many cases $Q=T$, but this is not true in general. To clarify this issue, we introduce the notion of quasi-isomorphic irreducible $T$-modules. We show that the…
Let $Γ$ denote a bipartite distance-regular graph with diameter $D \ge 4$ and valency $k \ge 3$. Let $X$ denote the vertex set of $Γ$, and let $A$ denote the adjacency matrix of $Γ$. For $x \in X$ let $T=T(x)$ denote the subalgebra of Mat$_X(\mathbb{C}$ generated by $A, E*_0, E*s_1, \ldots, E*_D$, where for $0 \le i \le D$, $E*_i$ represents the projection onto the $i$th subconstituent of $Γ$ with respect to $x$. We refer to $T$ as the {\em Terwilliger algebra} of $Γ$ with respect to $x$. An irreducible $T$-module $W$ is said to be {\em thin} whenever dim $E*_i W \le 1$ for $0 \le i \le D$. By…
In this paper we strengthen a result due to Li by showing that the third bi-power of a locally finite connected bipartite graph that admits a perfect matching is Hamilton-laceable, i.e. any two vertices from different bipartition classes are endpoints of some common Hamilton arc.
On quadratic embeddability of bipartite graphs and theta graphs
本文计算了去掉不相交边的完全二部图的二次嵌入常数，并研究了theta图的二次嵌入性质，最终构造出一类无限多的非二次可嵌入原始图。
An equitable partition for the distance-regular graph of the bilinear forms
本文研究双线性形式图的距离正则图的公平划分，提出(x,y)-划分，并将其特征向量作为T-模块的基础，分解为不可约T-模块的正交直和。
A problem of Erdős and Hajnal on paths with equal-degree endpoints
本文证明了 Erdős 和 Hajnal 提出的关于图中等度端点路径的问题，给出了边数下界，并推广到偶数阶图及相关极值问题。
A complement of the Erdős-Hajnal problem on paths with equal-degree endpoints
本研究提出一种处理大等度图的新方法，将Erdős-Hajnal等度端点路径问题的成立范围从n≥600拓展至所有n≥2，完全解决了该问题。
Paths of length five with equal-degree endpoints
研究证明，对于n≥11，K_{n,n+1}是唯一具有至少n²+n条边且不含长度为五的等度端点路径的2n+1顶点图，扩展了Chen和Ma的猜想。
Maximum Edge Open Packing on AT-Free, Chordal, and Convex Bipartite Graphs
针对AT-free图、弦图和凸二分图，分别提出基于星冲突图、团树分解和边界索引的最大边开打包算法，时间复杂度分别为$O(n^2+m^4)$、$O(n^4)$和$O(n^4)$，并扩展了双凸二分图的结果。
Let $Y$ denote a $D$-class symmetric association scheme with $D \geq 3$, and suppose $Y$ is almost-bipartite P- and Q-polynomial. Let $x$ denote a vertex of $Y$ and let $T=T(x)$ denote the corresponding Terwilliger algebra. We prove that any irreducible $T$-module $W$ is both thin and dual thin in the sense of Terwilliger. We produce two bases for $W$ and describe the action of $T$ on these bases. We prove that the isomorphism class of $W$ as a $T$-module is determined by two parameters, the dual endpoint and diameter of $W$. We find a recurrence which gives the multiplicities with which the i…
Paths of even length with equal-degree endpoints
研究对于固定整数\(\ell\geq2\)和足够大的\(n\)，满足特定条件的\(2n\)顶点图的问题，核心方法未提及，主要贡献是证明该图为半图\(H_n\)，解决相关问题。
We address the following question: When a randomly chosen regular bipartite multi--graph is drawn in the plane in the ``standard way'', what is the distribution of its maximum size planar matching (set of non--crossing disjoint edges) and maximum size planar subgraph (set of non--crossing edges which may share endpoints)? The problem is a generalization of the Longest Increasing Sequence (LIS) problem (also called Ulam's problem). We present combinatorial identities which relate the number of $r$-regular bipartite multi--graphs with maximum planar matching (maximum planar subgraph)of at most $…
We address the following question: When a randomly chosen regular bipartite multi--graph is drawn in the plane in the ``standard way'', what is the distribution of its maximum size planar matching (set of non--crossing disjoint edges) and maximum size planar subgraph (set of non--crossing edges which may share endpoints)? The problem is a generalization of the Longest Increasing Sequence (LIS) problem (also called Ulam's problem). We present combinatorial identities which relate the number of r-regular bipartite multi--graphs with maximum planar matching (maximum planar subgraph) of at most d …
Paired $(n-1)$-to-$(n-1)$ disjoint path covers in bipartite transposition-like graphs
本文引入由低秩图归纳构造的二部置换类图，证明所有$n$阶该类图在端点集$S$、$T$分属不同划分集时，均存在配对$(n-1)$-到-$(n-1)$不交路覆盖。
A graph $\mathcal{H}=(W,E_\mathcal{H})$ is said to have {\em bandwidth} at most $b$ if there exists a labeling of $W$ as $w_1,w_2,\dots,w_n$ such that $|i-j|\leq b$ for every edge $w_iw_j\in E_\mathcal{H}$. We say that $\mathcal{H}$ is a {\em balanced $(β,Δ)$-graph} if it is a bipartite graph with bandwidth at most $β|W|$ and maximum degree at most $Δ$, and it also has a proper 2-coloring $χ:W\rightarrow[2]$ such that $||χ^{-1}(1)|-|χ^{-1}(2)||\leqβ|χ^{-1}(2)|$. In this paper, we prove that for every $γ>0$ and every natural number $Δ$, there exists a constant $β>0$ such that for every balanced…
A Sharp Spectral Mantel Theorem for Quantum Transport
该研究针对量子输运相关的无三角图极值谱问题，确定了特定参数区间内图函子的最大值及唯一最大化子，给出了稳定性结果，相关有限极值值收敛到图函子最大值。
Vertex-edge domination on subclasses of bipartite graphs
本文证明了星凸和梳凸双图中顶点-边支配集问题仍为NP难，并改进了凸双图的算法以正确计算最优解。
Algorithms for the Maximum Edge Open Packing Problem
研究最大边开打包问题的计算复杂性，在距离遗传图和双凸二分图上给出多项式时间算法，并在弦图上给出参数化算法。
$k$-Connected Subgraphs of All Orders in Large Graphs with Minimum Degree at Least $n/q$
研究证实刘和宁猜想，即最小度至少为n/q的2-连通图含各阶2-连通子图。通过构造子图D及添加短路径，结合平均论证得到结果，还提出关于r-连通图的类似猜想。
Non-bipartite graphs without theta subgraphs
本文研究不含 theta 子图的非二部图的谱极值问题，证明了当 n 充分大时谱极图族包含于边数极图族，并完全刻画了这两类极图族中的所有图。
Edge-Number Bounds for the Inversion Diameter of Graphs
研究有向图反转图直径的边数界，证明上界为2√|E(G)|，下界为|E(G)|/|V(G)|，并改进二部图的上界。
Distance-regular graphs with classical parameters that support a uniform structure: case $q \ge 2$
本文研究具有经典参数的距离正则图在 $q \ge 2$ 时支持一致结构的条件，在端点1的不可约模为薄的假设下，证明参数需满足 $\alpha=0$ 或 $\alpha=q$、$\beta=q^2(q^D-1)/(q-1)$ 且 $D$ 为6的倍数。
A Tight Lower Bound on Cubic Vertices and Upper Bounds on Thin and Non-thin edges in Planar Braces
该研究证明不存在不含三次顶点的平面brace，得到了平面brace中三次顶点数的定量紧下界，以及细边与非细边的紧上界，推进了相关匹配覆盖图的边结构猜想研究。
Two Results on Outer-String Graphs
本文研究了外弦图的约束外弦表示和外1-弦表示的判定问题，证明了双色图及{C3,C5}自由图的约束外弦表示可多项式时间判定，并证明了外k-弦表示的判定为NP难问题。
Bichromatic Geometric Spanners
研究平面上双色点完全二分图的生成子图，针对边数与拉伸问题，给出边数为\(O(\sqrt{1/\varepsilon}\cdot n)\)的\((3 + \varepsilon)-\)生成子图新构造，解决多年开放问题并提出新问题，还研究实线上双色点情况，给出相关生成子图结论。
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