设G为一个有限简单图，B⊆V(G)。我们引入了双变量B限制的团多项式$$ C_B(G;x,y) = \sum_{\substack{K \subseteq V \ K \text{ 是一个团}}} x^{|K|} y^{|K \cap B|}, $$其中x^i y^j的系数计数大小为i且恰好有j个顶点在B中的团。该多项式同时捕捉了组合结构、局部极值性质以及与子集B相关的谱约束。首先，我们开发了顶点和边删除递推公式，推广了经典的团多项式结果。这些递推公式表明，在固定y∈[0,1]时，最大的负根ζ_G(B;y)在诱导子图和生成子图下具有单调性。从这里，我们推导了B-独立数、B-圈长以及限制于B的团密度的界限。接下来，我们证明对于任何整数r≥1，任何r连通的K_{r+3}-自由弦图G，以及任何子集B⊆V(G)，双变量团多项式C_B(G;x,y)是实稳定的。然后，我们将C_B(G;x,y)与谱图理论联系起来。对于(n,d,λ)-图，通过Tanner不等式限制团增长在B内，从而得到系数和ζ_G(B;y)的显式界限。最后，我们分析了该框架中的加权顶点和同态障碍，给出了一个一般的无同态准则。我们还以几个有趣的问题作为论文的结尾，为年轻和有动力的学者提供研究方向。
英文摘要：
Let $G$ be a finite simple graph and $B \subseteq V(G)$. We introduce the \emph{bivariate $B$-restricted clique polynomial} \[ C_B(G;x,y) = \sum_{\substack{K \subseteq V \\ K \text{ is a clique}}} x^{|K|} y^{|K \cap B|}, \] where the coefficient of $x^i y^j$ counts cliques of size $i$ with exactly $j$ vertices in $B$. This polynomial simultaneously captures combinatorial structure, local extremal properties, and spectral constraints associated with the subset $B$. \\ First, we develop vertex and edge deletion recurrences, generalizing classical clique polynomial results. These recurrences imply monotonicity for the largest negative root $ζ_G(B;y)$ (viewed as a polynomial in $x$ for fixed $y \in [0,1]$) under induced and spanning subgraphs.
From this, we derive bounds on $B$-independence numbers, $B$-girth, and clique densities restricted to $B$. \\ Next, we prove that for any integer $r \ge 1$, any $r$-connected $K_{r+3}$-free chordal graph $G$, and any subset $B \subseteq V(G)$, the bivariate clique polynomial $C_B(G;x,y)$ is real-stable. \\ Then, we connect $C_B(G;x,y)$ with spectral graph theory. For $(n,d,λ)$-graphs, expansion constraints via Tanner's inequality limit clique growth within $B$, yielding explicit bounds on coefficients and $ζ_G(B;y)$. \\ Finally, we analyze weighted vertices and homomorphism obstructions in this framework, giving a general no-homomorphism criterion. We also conclude the paper with a couple of interesting open problems for young and motivated researchers.
Let $G$ be a graph and $t\ge 0$. The largest reduced neighborhood clique cover number of $G$, denoted by ${\hatβ}_t(G)$, is the largest, overall $t$-shallow minors $H$ of $G$, of the smallest number of cliques that can cover any closed neighborhood of a vertex in $H$. It is known that ${\hatβ}_t(G)\le s_t$, where $G$ is an incomparability graph and $s_t$ is the number of leaves in a largest $t-$shallow minor which is isomorphic to an induced star on $s_t$ leaves. In this paper we give an overview of the properties of ${\hatβ}_t(G)$ including the connections to the greatest reduced average dens…
Let $G$ be a graph and $t\ge 0$. A new graph parameter termed the largest reduced neighborhood clique cover number of $G$, denoted by ${\hatβ}_t(G)$, is introduced. Specifically, ${\hatβ}_t(G)$ is the largest, overall $t$-shallow minors $H$ of $G$, of the smallest number of cliques that can cover any closed neighborhood of a vertex in $H$. We verify that ${\hatβ}_t(G)=1$ when $G$ is chordal, and, ${\hatβ}_t(G)\le s$, where $G$ is an incomparability graph that does not have a $t-$shallow minor which is isomorphic to an induced star on $s$ leaves. Moreover, general properties of ${\hatβ}_t(G)$ i…
It is shown that any graph with maximum degree $Δ$ in which the average degree of the induced subgraph on the set of all neighbors of any vertex exceeds $\frac{6k^2}{6k^2 + 1}Δ+ k + 6$ is either $(Δ- k)$-colorable or contains a clique on more than $Δ- 2k$ vertices. In the $k=1$ case we improve the bound on the average degree to $\frac23Δ+ 4$ and the bound on the clique number to $Δ-1$. As corollaries, we show that every graph satisfies $χ\leq \max\set{ω, Δ- 1, 4α}$ and every graph satisfies $χ\leq \max\set{ω, Δ- 1, \ceil{\frac{15 + \sqrt{48n + 73}}{4}}}$.
An identifying code in a graph $G$ is a dominating set $C$ such that the closed neighborhood of each vertex in $G$ has a distinct intersection with $C$. In 2008, Gravier et al. determined the minimum cardinality of an identifying code of the Cartesian product of two cliques with the same size. In this note, we generalize this result to the Cartesian product of any two nontrivial cliques.
An identifying code in a graph is a dominating set that also has the property that the closed neighborhood of each vertex in the graph has a distinct intersection with the set. It was recently shown by Gravier, Moncel and Semri that the minimum cardinality of an identifying code for the Cartesian product of two cliques of the same order n is the floor of 3n/2. We consider identifying codes of the direct product of two cliques. In particular, we answer a question of Klavzar and determine the minimum cardinality of an identifying code for the direct product of any two cliques.
In the way of proving Kneser's conjecture, László Lovász settled out a new lower bound for the chromatic number. He showed that if neighborhood complex $\mathcal{N}(G)$ of a graph $G$ is topologically $k$-connected, then its chromatic number is at least $k+3$. Then he completed his proof by showing that this bound is tight for the Kneser graph. However, there are some graphs where this bound is not useful at all, even comparing by the obvious bound; the clique number. For instance, if a graph contains no complete bipartite graph $\mathcal{K}_{l, m}$, then its Lovász's bound is at most $l+m-1$.…
The greedy coloring algorithm shows that a graph of maximum degree at most $Δ$ has chromatic number at most $Δ+ 1$, and this is tight for cliques. Much attention has been devoted to improving this "greedy bound" for graphs without large cliques. Brooks famously proved that this bound can be improved by one if $Δ\geq 3$ and the graph contains no clique of size $Δ+ 1$. Reed's Conjecture states that the "greedy bound" can be improved by $k$ if the graph contains no clique of size $Δ+ 1 - 2k$. Johansson proved that the "greedy bound" can be improved by a factor of $Ω(\ln(Δ)^{-1})$ or $Ω\left(\frac…
A graph is called $P_t$-free} if it does not contain a $t$-vertex path as an induced subgraph. While $P_4$-free graphs are exactly cographs, the structure of $P_t$-free graphs for $t \geq 5$ remains little understood. On one hand, classic computational problems such as Maximum Weight Independent Set (MWIS) and $3$-Coloring are not known to be NP-hard on $P_t$-free graphs for any fixed $t$. On the other hand, despite significant effort, polynomial-time algorithms for MWIS in $P_6$-free graphs~[SODA 2019] and $3$-Coloring in $P_7$-free graphs~[Combinatorica 2018] have been found only recently. I…
For a graph G, the neighborhood complex N[G] is the simplicial complex having all subsets of vertices with a common neighbor as its faces. It is a well known result of Lovasz that if N[G] is k-connected, then the chromatic number of G is at least k + 3. We prove that the connectivity of the neighborhood complex of a random graph is tightly concentrated, almost always between 1/2 and 2/3 of the expected clique number. We also show that the number of dimensions of nontrivial homology is almost always small, O(log d), compared to the expected dimension d of the complex itself.
For graphs $F$ and $H$, we say $F$ is Ramsey for $H$ if every $2$-coloring of the edges of $F$ contains a monochromatic copy of $H$. The graph $F$ is Ramsey $H$-minimal if $F$ is Ramsey for $H$ and there is no proper subgraph $F'$ of $F$ so that $F'$ is Ramsey for $H$. Burr, Erdos, and Lovasz defined $s(H)$ to be the minimum degree of $F$ over all Ramsey $H$-minimal graphs $F$. Define $H_{t,d}$ to be a graph on $t+1$ vertices consisting of a complete graph on $t$ vertices and one additional vertex of degree $d$. We show that $s(H_{t,d})=d^2$ for all values $1<d\le t$; it was previously known t…
Let $G=(V,E)$ be a graph. A $k$-neighborhood in $G$ is a set of vertices consisting of all the vertices at distance at most $k$ from some vertex of $G$. The hypergraph on vertex set $V$ which edge set consists of all the $k$-neighborhoods of $G$ for all $k$ is the neighborhood hypergraph of $G$. Our goal in this paper is to investigate the complexity of a graph in terms of its neighborhoods. Precisely, we define the distance VC-dimension of a graph $G$ as the maximum taken over all induced subgraphs $G'$ of $G$ of the VC-dimension of the neighborhood hypergraph of $G'$. For a class of graphs, …
Inspired by the study of community structure in connection networks, we introduce the graph polynomial $Q(G;x,y)$, the bivariate generating function which counts the number of connected components in induced subgraphs. We give a recursive definition of $Q(G;x,y)$ using vertex deletion, vertex contraction and deletion of a vertex together with its neighborhood and prove a universality property. We relate $Q(G;x,y)$ to other known graph invariants and graph polynomials, among them partition functions, the Tutte polynomial, the independence and matching polynomials, and the universal edge elimina…
The family of $\mathfrak{I}$-contractible graphs and contractible transformations was defined by A. Ivashchenko in the mid-90's. In this paper we study the collapsibility and homological properties of the clique complex associated to $\mathfrak{I}$-contractible graphs. We show that for any graph in a special subfamily of the $\mathfrak{I}$-contractible graphs (the strong $\mathfrak{I}$-contractible ones) its clique complex is collapsible. Moreover, we present an algorithm that allows us to verify if any graph is strong $\mathfrak{I}$-contractible, as well as an algorithm to delete those vertic…
The Vapnik-Chervonenkis dimension (in short, VC-dimension) of a graph is defined as the VC-dimension of the set system induced by the neighborhoods of its vertices. We show that every $n$-vertex graph with bounded VC-dimension contains a clique or an independent set of size at least $e^{(\log n)^{1 - o(1)}}$. The dependence on the VC-dimension is hidden in the $o(1)$ term. This improves the general lower bound, $e^{c\sqrt{\log n}}$, due to Erdos and Hajnal, which is valid in the class of graphs satisfying any fixed nontrivial hereditary property. Our result is almost optimal and nearly matches…
Time-Optimal APSP and Matrix Multiplication in Classes of Linear Neighborhood Complexity
CNRS, ENS de Lyon, Université Claude Bernard Lyon 1, LIP UMR 5668（法国国家科学研究中心、里昂高等师范学校、克莱德·贝尔纳·里昂第一大学、LIP联合研究单位）；Institute of Informatics, University of Warsaw（华沙大学信息学研究所）；Max Planck Institute for Informatics, Saarland Informatics Campus（马克斯·普朗克 informatics 研究所、萨尔兰 informatics 校园）
We study the connections between the notions of combinatorial discrepancy and graph degeneracy. In particular, we prove that the maximum discrepancy over all subgraphs $H$ of a graph $G$ of the neighborhood set system of $H$ is sandwiched between $Ω(\log\mathrm{deg}(G))$ and $\mathcal{O}(\mathrm{deg}(G))$, where $\mathrm{deg}(G)$ denotes the degeneracy of $G$. We extend this result to inequalities relating weak coloring numbers and discrepancy of graph powers and deduce a new characterization of bounded expansion classes. Then, we switch to a model theoretical point of view, introduce pointer …
This paper deals with the partition function of the Ising model from statistical mechanics, which is used to study phase transitions in physical systems. A special case of interest is that of the Ising model with constant energies and external field. One may consider such an Ising system as a simple graph together with vertex and edge weights. When these weights are considered indeterminates, the partition function for the constant case is a trivariate polynomial Z(G;x,y,z). This polynomial was studied with respect to its approximability by L. A. Goldberg, M. Jerrum and M. Paterson in 2003. Z(…
