We give a new bound on the parameter $λ$ (number of common neighbors of a pair of adjacent vertices) in a distance-regular graph $G$, improving and generalizing bounds for strongly regular graphs by Spielman (1996) and Pyber (2014). The new bound is one of the ingredients of recent progress on the complexity of testing isomorphism of strongly regular graphs (Babai, Chen, Sun, Teng, Wilmes 2013). The proof is based on a clique geometry found by Metsch (1991) under certain constraints on the parameters. We also give a simplified proof of the following asymptotic consequence of Metsch's result: if $kμ= o(λ^2)$ then each edge of $G$ belongs to a unique maximal clique of size asymptotically equal to $λ$, and all other cliques have size $o(λ)$. Here $k$ denotes the degree and $μ$ the number of common neighbors of a pair of vertices at distance 2. We point out that Metsch's cliques are "asymptotically Delsarte" when $kμ= o(λ^2)$, so families of distance-regular graphs with parameters satisfying $kμ= o(λ^2)$ are "asymptotically Delsarte-geometric."
In this paper, we study the order of a maximal clique in an amply regular graph with a fixed smallest eigenvalue by considering a vertex that is adjacent to some (but not all) vertices of the maximal clique. As a consequence, we show that if a strongly regular graph contains a Delsarte clique, then the parameter $μ$ is either small or large. Furthermore, we obtain a cubic polynomial that assures that a maximal clique in an amply regular graph is either small or large (under certain assumptions). Combining this cubic polynomial with the claw-bound, we rule out an infinite family of feasible par…
We determine new upper bounds for the clique numbers of strongly regular graphs in terms of their parameters. These bounds improve on the Delsarte bound for infinitely many feasible parameter tuples for strongly regular graphs, including infinitely many parameter tuples that correspond to Paley graphs.
The $k$-th $p$-power of a graph $G$ is the graph on the vertex set $V(G)^k$, where two $k$-tuples are adjacent iff the number of their coordinates which are adjacent in $G$ is not congruent to 0 modulo $p$. The clique number of powers of $G$ is poly-logarithmic in the number of vertices, thus graphs with small independence numbers in their $p$-powers do not contain large homogenous subsets. We provide algebraic upper bounds for the asymptotic behavior of independence numbers of such powers, settling a conjecture of Alon and Lubetzky up to a factor of 2. For precise bounds on some graphs, we ap…
We investigate strongly regular graphs for which Hoffman's ratio bound and Cvetcović's inertia bound are equal. This means that $ve^- = m^-(e^- - k)$, where $v$ is the number of vertices, $k$ is the regularity, $e^-$ is the smallest eigenvalue, and $m^-$ is the multiplicity of $e^-$. We show that Delsarte cocliques do not exist for all Taylor's $2$-graphs and for point graphs of generalized quadrangles of order $(q,q^2-q)$ for infinitely many $q$. For cases where equality may hold, we show that for nearly all parameter sets, there are at most two Delsarte cocliques.
A non-complete geometric distance-regular graph is the point graph of a partial geometry in which the set of lines is a set of Delsarte cliques. In this paper, we prove that for fixed integer $m\geq 2$, there are only finitely many non-geometric distance-regular graphs with smallest eigenvalue at least $-m$, diameter at least three and intersection number $c_2 \geq 2$.
A non-complete \drg $Γ$ is called geometric if there exists a set $\mathcal{C}$ of Delsarte cliques such that each edge of $Γ$ lies in a unique clique in $\mathcal{C}$. In this paper, we determine the non-complete distance-regular graphs satisfying $\max \{3, 8/3}(a_1+1)\}<k<4a_1+10-6c_2$. To prove this result, we first show by considering non-existence of 4-claws that any non-complete distance-regular graph satisfying $\max \{3, \8/3}(a_1+1)\}<k<4a_1+10-6c_2$ is a geometric \drg with smallest eigenvalue -3. Moreover, we classify the geometric \drg s with smallest eigenvalue -3. As an applicat…
We exhibit infinitely many examples of edge-regular graphs that have regular cliques and that are not strongly regular. This answers a question of Neumaier from 1981.
We exhibit a new construction of edge-regular graphs with regular cliques that are not strongly regular. The infinite family of graphs resulting from this construction includes an edge-regular graph with parameters $(24,8,2)$. We also show that edge-regular graphs with $1$-regular cliques that are not strongly regular must have at least $24$ vertices.
In this paper we consider $r$-regular graphs $G$ that admit the vertex set partition such that one of the induced subgraphs is the join of an $s$-vertex clique and a $t$-vertex co-clique and represents a star complement for an eigenvalue $μ$ of $G$. The cases in which one of the parameters $s, t$ is less than 2 or $μ=r$ are already resolved. It is conjectured in [J. Wang, X. Yuan, L. Liu, Regular graphs with a prescribed complete multipartite graph as a star complement, Linear Algebra Appl.~579 (2019) 302--319] that if $s, t\geq 2$ and $μ\neq r$, then $μ=-2, t=2$ and $G=\overline{(s+1)K_2}$. F…
A regular clique in a regular graph is a clique such that every vertex outside of the clique is adjacent to the same positive number of vertices inside the clique. We continue the study of regular cliques in edge-regular graphs initiated by A. Neumaier in the 1980s and attracting current interest. We thus define a Neumaier graph to be an non-complete edge-regular graph containing a regular clique, and a strictly Neumaier graph to be a non-strongly regular Neumaier graph. We first prove some general results on Neumaier graphs and their feasible parameter tuples. We then apply these results to d…
A collection of sets is intersecting, if any pair of sets in the collection has nonempty intersection. A collection of sets \(\mathcal{C}\) has the Helly property if any intersecting subcollection has nonempty intersection. A graph is /Helly/ if the collection of maximal complete subgraphs of \(G\) has the Helly property. We prove that if \(G\) is a \(k\)-regular graph with \(n\) vertices such that \(n>3k+\sqrt{2k^{2}-k}\), then the complement \(\bar{G}\) is not Helly. We also consider the problem of whether the properties of Hellyness and convergence under the clique graph operator are equiva…
A regular graph is co-edge regular if there exists a constant $μ$ such that any two distinct and non-adjacent vertices have exactly $μ$ common neighbors. In this paper, we show that for integers $s\ge 2$ and $n$ large enough, any co-edge-regular graph which is cospectral with the $s$-clique extension of the triangular graph $T((n)$ is exactly the $s$-clique extension of the triangular graph $T(n)$.
In this paper we show that for integers $s\geq2$, $t\geq1$, any co-edge-regular graph which is cospectral with the $s$-clique extension of the $t\times t$-grid is the $s$-clique extension of the $t\times t$-grid, if $t$ is large enough. Gavrilyuk and Koolen used a weaker version of this result to show that the Grassmann graph $J_q(2D,D)$ is characterized by its intersection array as a distance-regular graph, if $D$ is large enough.
We prove that a random $d$-regular graph, with high probability, is a cut sparsifier of the clique with approximation error at most $\left(2\sqrt{\frac 2 π} + o_{n,d}(1)\right)/\sqrt d$, where $2\sqrt{\frac 2 π} = 1.595\ldots$ and $o_{n,d}(1)$ denotes an error term that depends on $n$ and $d$ and goes to zero if we first take the limit $n\rightarrow \infty$ and then the limit $d \rightarrow \infty$. This is established by analyzing linear-size cuts using techniques of Jagannath and Sen derived from ideas in statistical physics, and analyzing small cuts via martingale inequalities. We also prov…
We consider the orthogonality graph Omega(n) with 2^n vertices corresponding to the 0-1 n-vectors, two vertices adjacent if and only if the Hamming distance between them is n/2. We show that the stability number of Omega(16) is alpha(Omega(16))= 2304, thus proving a conjecture by Galliard. The main tool we employ is a recent semidefinite programming relaxation for minimal distance binary codes due to Schrijver. As well, we give a general condition for Delsarte bound on the (co)cliques in graphs of relations of association schemes to coincide with the ratio bound, and use it to show that for Om…
Write f(n) for the minimum number of 4-cliques in graphs of order n with triangle-free complement. Finding f(n) is a particular case of a problem raised by Erdos in 1962. We give an upper bound of f(n) and a matching lower bound when the graph is close to regular.
We give lower bounds on the size and total size of clique partitions of a graph in terms of its spectral radius and minimum degree, and derive a spectral upper bound for the maximum number of edge-disjoint $t$-cliques. The extremal graphs attaining the bounds are exactly the block graphs of Steiner $2$-designs and the regular graphs with $K_t$-decompositions, respectively.
A graph $G$ is called $C_4$-free if it does not contain the cycle $C_4$ as an induced subgraph. Hubenko, Solymosi and the first author proved (answering a question of Erd\H os) a peculiar property of $C_4$-free graphs: $C_4$ graphs with $n$ vertices and average degree at least $cn$ contain a complete subgraph (clique) of size at least $c'n$ (with $c'= 0.1c^2n$). We prove here better bounds (${c^2n\over 2+c}$ in general and $(c-1/3)n$ when $ c \le 0.733$) from the stronger assumption that the $C_4$-free graphs have minimum degree at least $cn$. Our main result is a theorem for regular graphs, c…
Delsarte, Goethals, and Seidel (1977) used the linear programming method in order to find bounds for the size of spherical codes endowed with prescribed inner products between distinct points in the code. In this paper, we develop the linear programming method to obtain bounds for the number of vertices of connected regular graphs endowed with given distinct eigenvalues. This method is proved by some "dual" technique of the spherical case, motivated from the theory of association scheme. As an application of this bound, we prove that a connected $k$-regular graph satisfying $g>2d-1$ has the mi…
In this paper we introduce the concept of clique disjoint edge sets in graphs. Then, for a graph $G$, we define the invariant $η(G)$ as the maximum size of a clique disjoint edge set in $G$. We show that the regularity of the binomial edge ideal of $G$ is bounded above by $η(G)$. This, in particular, settles a conjecture on the regularity of binomial edge ideals in full generality.
We study structural properties of graphs with fixed clique number and high minimum degree. In particular, we show that there exists a function $L=L(r,\varepsilon)$, such that every $K_r$-free graph $G$ on $n$ vertices with minimum degree at least $(\frac{2r-5}{2r-3}+\varepsilon)n$ is homomorphic to a $K_r$-free graph on at most $L$ vertices. It is known that the required minimum degree condition is approximately best possible for this result. For $r=3$ this result was obtained by Łuczak [On the structure of triangle-free graphs of large minimum degree, Combinatorica 26 (2006), no. 4, 489-493] …
Many regular graphs admit a natural partition of their edge set into cliques of the same order such that each vertex is contained in the same number of cliques. In this paper, we study the mixing rate of certain random walks on such graphs and we generalize previous results of Alon, Benjamini, Lubetzky and Sodin regarding the mixing rates of non-backtracking random walks on regular graphs.
This paper gives a tight upper bound on the spectral radius of the signless Laplacian of graphs of given order and clique number. More precisely, let G be a graph of order n, let A be its adjacency matrix, and let D be the diagonal matrix of the row-sums of A. If G has clique number r, then the largest eigenvalue q(G) of the matrix Q=A+D satisfies q(G)<= 2(1-1/r)n. If G is a complete regular r-partite graph, then equality holds in the above inequality. This result confirms a conjecture of Hansen and Lucas.
Let $R$ be a commutative ring with non-zero identity. The cozero-divisor graph of $R$, denoted by $Γ^{\prime}(R)$, is a graph with vertices in $W^*(R)$, which is the set of all non-zero and non-unit elements of $R$, and two distinct vertices $a$ and $b$ in $W^*(R)$ are adjacent if and only if $a\not\in Rb$ and $b\not\in Ra$. In this paper, we show that the cozero-divisor graph of a von Neumann regular ring with finite clique number is not only weakly perfect but also perfect. Also, an explicit formula for the clique number is given.
A strongly regular graph is called trivial if it or its complement is a union of disjoint cliques. We prove that every infinite family of nontrivial strongly regular graphs is quasi-random in the sense of Chung, Graham and Wilson.
In 1999, De Simone and Körner conjectured that every graph without induced $C_5,C_7,\overline{C}_7$ contains a clique cover $\mathcal C$ and a stable set cover $\mathcal I$ such that every clique in $\mathcal C$ and every stable set in $\mathcal I$ have a vertex in common. This conjecture has roots in information theory and became known as the Normal Graph Conjecture. Here we prove that all graphs of bounded maximum degree and sufficiently large odd girth (linear in the maximum degree) are normal. This implies that for every fixed $d$, random $d$-regular graphs are a.a.s. normal.
We prove that the clique graph operator $k$ is divergent on a locally cyclic graph $G$ (i.e. $N_G(v)$ is a circle) with minimum degree $δ(G)=6$ if and only if $G$ is $6$-regular. The clique graph $kG$ of a graph $G$ has the maximal complete subgraphs of $G$ as vertices, and the edges are given by non-empty intersections. If all iterated clique graphs of $G$ are pairwise non-isomorphic, the graph $G$ is $k$-divergent; otherwise, it is $k$-convergent. To prove our claim, we explicitly construct the iterated clique graphs of those infinite locally cyclic graphs with $δ\geq6$ which induce simply c…
Moderately beyond clique-width: reduced component max-leaf and related parameters
Univ Lyon, CNRS, ENS de Lyon, Université Claude Bernard Lyon 1（里昂大学、法国国家科学研究中心、巴黎高等师范学院里昂分校、克洛德·贝尔纳里昂大学）；Institute of Informatics, University of Warsaw（华沙大学信息学研究所）；Department of Mathematics and Research Institute for Natural Sciences, Hanyang University（汉阳大学数学系与自然科学研究所）；Discrete Mathematics Group, Institute for Basic Science (IBS)（基础科学研究院离散数学组）
The clique complex of a graph G is a simplicial complex whose simplices are all the cliques of G, and the line graph L(G) of G is a graph whose vertices are the edges of G and the edges of L(G) are incident edges of G. In this article, we determine the homotopy type of the clique complexes of line graphs for several classes of graphs including triangle-free graphs, chordal graphs, complete multipartite graphs, wheel-free graphs, and 4-regular circulant graphs. We also give a closed form formula for the homotopy type of these complexes in several cases.
A $t$-walk-regular graph is a graph for which the number of walks of given length between two vertices depends only on the distance between these two vertices, as long as this distance is at most $t$. Such graphs generalize distance-regular graphs and $t$-arc-transitive graphs. In this paper, we will focus on 1- and in particular 2-walk-regular graphs, and study analogues of certain results that are important for distance regular graphs. We will generalize Delsarte's clique bound to 1-walk-regular graphs, Godsil's multiplicity bound and Terwilliger's analysis of the local structure to 2-walk-r…
The cube graph Q_n is the skeleton of the n-dimensional cube. It is an n-regular graph on 2^n vertices. The Ramsey number r(Q_n, K_s) is the minimum N such that every graph of order N contains the cube graph Q_n or an independent set of order s. Burr and Erdos in 1983 asked whether the simple lower bound r(Q_n, K_s) >= (s-1)(2^n - 1)+1 is tight for s fixed and n sufficiently large. We make progress on this problem, obtaining the first upper bound which is within a constant factor of the lower bound.
If C is a clutter with n vertices and q edges whose clutter matrix has column vectors V={v1,...,vq}, we call C an Ehrhart clutter if {(v1,1),...,(vq,1)} is a Hilbert basis. Letting A(P) be the Ehrhart ring of P=conv(V), we are able to show that if A is the clutter matrix of a uniform, unmixed MFMC clutter C, then C is an Ehrhart clutter and in this case we provide sharp bounds on the Castelnuovo-Mumford regularity of A(P). Motivated by the Conforti-Cornuejols conjecture on packing problems, we conjecture that if C is both ideal and the clique clutter of a perfect graph, then C has the MFMC pro…
The regular graph of ideals of the commutative ring $R$, denoted by $Γ_{reg}(R)$, is a graph whose vertex set is the set of all non-trivial ideals of $R$ and two distinct vertices $I$ and $J$ are adjacent if and only if either $I$ contains a $J$-regular element or $J$ contains an $I$-regular element. In this paper, it is shown that for every Artinian ring $R$, the edge chromatic number of $Γ_{reg}(R)$ equals its maximum degree. Then a formula for the clique number of $Γ_{reg}(R)$ is given. Also, it is proved that for every reduced ring $R$ with $n(\geq3)$ minimal prime ideals, the edge chromat…
We prove that, for any $t\ge 3$, there exists a constant $c=c(t)>0$ such that any $d$-regular $n$-vertex graph with the second largest eigenvalue in absolute value~$λ$ satisfying $λ\le c d^{t-1}/n^{t-2}$ contains vertex-disjoint copies of $K_t$ covering all but at most $n^{1-1/(8t^4)}$ vertices. This provides further support for the conjecture of Krivelevich, Sudakov and Szábo [\emph{Triangle factors in sparse pseudo-random graphs}, Combinatorica \textbf{24} (2004), pp.~403--426] that $(n,d,λ)$-graphs with $n\in 3\mathbb{N}$ and $λ\leq cd^{2}/n$ for a suitably small absolute constant~$c>0$ con…
In a sequence of four papers, we prove the following results (via a unified approach) for all sufficiently large $n$: (i) [1-factorization conjecture] Suppose that $n$ is even and $D\geq 2\lceil n/4\rceil -1$. Then every $D$-regular graph $G$ on $n$ vertices has a decomposition into perfect matchings. Equivalently, $χ'(G)=D$. (ii) [Hamilton decomposition conjecture] Suppose that $D \ge \lfloor n/2 \rfloor $. Then every $D$-regular graph $G$ on $n$ vertices has a decomposition into Hamilton cycles and at most one perfect matching. (iii) We prove an optimal result on the number of edge-disjoint …
We prove that there is no strongly regular graph (SRG) with parameters (460,153,32,60). The proof is based on a recent lower bound on the number of 4-cliques in a SRG and some applications of Euclidean representation of SRGs.
