Understanding the structure of a graph along with the structure of its subgraphs is important for several problems in graph theory. Two examples are the Reconstruction Conjecture and isomorph-free generation. This paper raises the question of which pairs of groups can be represented as the automorphism groups of a graph and a vertex-deleted subgraph. This, and more surprisingly the analogous question for edge-deleted subgraphs, are answered in the most positive sense using concrete constructions.
Let $k$, $m$ and $r$ be three integers such that $2\leq k\leq m\leq r$. Let $G$ be a $2r$-regular, $2m$-edge-connected graph of odd order. We obtain some sufficient conditions, which guarantee $G-v$ contains a $k$-factor for all $v\in V(G)$.
Any finite group can be encoded as the automorphism group of an unlabeled simple graph. Recently Hartke, Kolb, Nishikawa, and Stolee (2010) demonstrated a construction that allows any ordered pair of finite groups to be represented as the automorphism group of a graph and a vertex-deleted subgraph. In this note, we describe a generalized scenario as a game between a player and an adversary: An adversary provides a list of finite groups and a number of rounds. The player constructs a graph with automorphism group isomorphic to the first group. In the following rounds, the adversary selects a gr…
A b-coloring of a graph is a proper coloring of its vertices such that each color class contains a vertex adjacent to at least one vertex of every other color class. The b-chromatic number of a graph is the largest integer k such that the graph has a b-coloring with k colors. In this work we present lower bounds for the b-chromatic number of a vertex-deleted subgraph of a graph, particularly regarding two important classes, quasi-line and chordal graphs. We also get bounds for the b-chromatic number of G -{x}, when G is a graph with large girth.
We consider the Grassmann graph of $k$-dimensional subspaces of an $n$-dimensional vector space over the $q$-element field and its subgraph $Γ(n,k)_q$ formed by non-degenerate linear $[n,k]_q$ codes. We assume that $1<k<n-1$. It is well-known that every automorphism of the Grassmann graph is induced by a semilinear automorphism of the corresponding vector space or a semilinear isomorphism to the dual vector space; the second possibility is realized only if $n=2k$. Our results are the following: if $q\ge 3$ or $k\ne 2$, then every isomorphism of $Γ(n,k)_{q}$ to a subgraph of the Grassmann graph…
A graph is called a chain graph if it is bipartite and the neighborhoods of the vertices in each color class form a chain with respect to inclusion. Alazemi, Andelić and Simić conjectured that no chain graph shares a non-zero (adjacency) eigenvalue with its vertex-deleted subgraphs. We disprove this conjecture. However, we show that the assertion holds for subgraphs obtained by deleting vertices of maximum degrees in either of color classes. We also give a simple proof for the fact that chain graphs have no eigenvalue in the interval $(0,1/2)$.
Many degree sequences can only be realised in graphs that contain a `ds-completable card', defined as a vertex-deleted subgraph in which the erstwhile neighbours of the deleted vertex can be identified from their degrees, if one knows the degree sequence of the original graph. We obtain conditions on the degree sequence, such that graphs whose degree sequence satisfies one of the conditions must contain such a card. The methods allow all such sequences on graphs of order up to 10 to be identified, and some fraction of the sequences for larger graphs. Among other applications, this can be used …
Let G be a finite simple graph with automorphism group A(G). Then a spanning subgraph U of G is a fixing subgraph of G if G contains exactly $| A(G)|/ | A(G) \cap A(U)| $ subgraphs isomorphic to U: the graph G must always contain at least this number. If in addition $A(U) \subseteq A(G)$ then U is a strong fixing subgraph. Fixing subgraphs are important in many areas of graph theory. We consider them in the context of Hamiltonian graphs
A partial automorphism of a finite graph is an isomorphism between its vertex induced subgraphs. The set of all partial automorphisms of a given finite graph forms an inverse monoid under composition (of partial maps). We describe the algebraic structure of such inverse monoids by the means of the standard tools of inverse semigroup theory, namely Green's relations and some properties of the natural partial order, and give a characterization of inverse monoids which arise as inverse monoids of partial graph automorphisms. We extend our results to digraphs and edge-colored digraphs as well.
An identifying code of a graph is a dominating set which uniquely determines all the vertices by their neighborhood within the code. Whereas graphs with large minimum degree have small domination number, this is not the case for the identifying code number (the size of a smallest identifying code), which indeed is not even a monotone parameter with respect to graph inclusion. We show that every graph $G$ with $n$ vertices, maximum degree $Δ=ω(1)$ and minimum degree $δ\geq c\logΔ$, for some constant $c>0$, contains a large spanning subgraph which admits an identifying code with size $O\left(\fr…
Let $G$ be a group of permutations acting on an $n$-vertex set $V$, and $X$ and $Y$ be two simple graphs on $V$. We say that $X$ and $Y$ are $G$-isomorphic if $Y$ belongs to the orbit of $X$ under the action of $G$. One can naturally generalize the reconstruction problems so that when $G$ is $S_n$, the symmetric group, we have the usual reconstruction problems. In this paper, we study $G$-edge reconstructibility of graphs. We prove some old and new results on edge reconstruction and reconstruction from end vertex deleted subgraphs.
The distinguishing number (index) $D(G)$ ($D'(G)$) of a graph $G$ is the least integer $d$ such that $G$ has an vertex labeling (edge labeling) with $d$ labels that is preserved only by a trivial automorphism. Let $G$ be a connected graph constructed from pairwise disjoint connected graphs $G_1,\ldots ,G_k$ by selecting a vertex of $G_1$, a vertex of $G_2$, and identify these two vertices. Then continue in this manner inductively. We say that $G$ is obtained by point-attaching from $G_1, \ldots ,G_k$ and that $G_i$'s are the primary subgraphs of $G$. In this paper, we consider some particular …
Consider the Grassmann graph of $k$-dimensional subspaces of an $n$-dimensional vector space over the $q$-element field, $1<k<n-1$. Every automorphism of this graph is induced by a semilinear automorphism of the corresponding vector space or a semilinear isomorphism to the dual vector space; the second possibility is realized only for $n=2k$. Let $Γ(n,k)_q$ be the subgraph of the Grassman graph formed by all non-degenerate linear $[n,k]_q$ codes. If $q\ge 3$ or $k\ge 3$, then every isomorphism of $Γ(n,k)_{q}$ to a subgraph of the Grassmann graph can be uniquely extended to an automorphism of t…
This thesis focuses on theoretical and algorithmic tools for determining the numbers of induced subgraphs in strongly regular graphs, SRGs, and on further applications of such numbers. We consider in more detail a restricted class of these graphs, specifically those with no triangles. In this special case, there are infinitely many feasible sets of parameters for SRGs. Despite this fact there are only seven known examples of such graphs. we develop an algorithm which produces linear equations describing various relations between numbers of induced subgraphs of orders $o$ and $o-1$ in a SRG. We…
Механико-математический факультет Московского государственного университета（莫斯科国立大学机械数学系）；Московский центр фундаментальной и прикладной математики（莫斯科基础与应用数学中心）
Let $Γ$ be the graph on the roots of the $E_8$ root system, where any two distinct vertices $e$ and $f$ are connected by an edge with color equal to the inner product of $e$ and $f$. For any set $c$ of colors, let $Γ_c$ be the subgraph of $Γ$ consisting of all the $240$ vertices, and all the edges whose color lies in $c$. We consider cliques, i.e., complete subgraphs, of $Γ$ that are either monochromatic, or of size at most $3$, or a maximal clique in $Γ_c$ for some color set $c$, or whose vertices are the vertices of a face of the $E_8$ root polytope. We prove that, apart from two exceptions,…
A graph $G$ on $m$ vertices is pancyclic if it contains cycles of length $l$, $3\leq l \leq m$ as subgraphs in $G$. The complete graph $K_{9}$ on 9 vertices with a cycle $C_{9}$ of length 9 deleted from $K_{9}$ is denoted by $(K_{9}-C_{9})$. In this paper, we prove that $(K_{9}-C_{9})^{n}$, the Cartesian product of $(K_{9}-C_{9})$ taken $n$ times, is pancyclic.
Reconstructing graphs and their connectivity using graphlets
Institute of Computer Science of the Czech Academy of Sciences（捷克科学院计算机研究所）；Computer Science Institute of Charles University, Faculty of Mathematics and Physics, Charles University（查理大学数学物理学院计算机科学研究所）；Universitat Politècnica de Catalunya - BarcelonaTech (UPC)（加泰罗尼亚理工大学）
This paper deals with the Cayley graph $\mathrm{Cay}(\mathrm{Sym}_n,T_n),$ where the generating set consists of all block transpositions. A motivation for the study of these particular Cayley graphs comes from current research in Bioinformatics. As the main result, we prove that Aut$(\mathrm{Cay}(\mathrm{Sym}_n,T_n))$ is the product of the left translation group by a dihedral group $\mathsf{D}_{n+1}$ of order $2(n+1)$. The proof uses several properties of the subgraph $Γ$ of $\mathrm{Cay}(\mathrm{Sym}_n,T_n)$ induced by the set $T_n$. In particular, $Γ$ is a $2(n-2)$-regular graph whose automo…
The paper answers the question posed in a joint paper by A. A. Klyachko and N. M. Luneva about the optimality of the estimate for the cost of symmetry in graphs. The original estimate says that if n vertices can be removed from a connected graph so that there is no connected subgraph of isomorphic $Γ$ left in it, then at most $n|V(Γ)|$ vertices that form a set invariant under all automorphisms of the graph so that the graph does not contain a subgraph isomorphic to $Γ$. We will prove that there exists a graph $Γ$ for which this estimate is not optimal.
A coloured graph is k-ultrahomogeneous if every isomorphism between two induced subgraphs of order at most k extends to an automorphism. A coloured graph is t-tuple regular if the number of vertices adjacent to every vertex in a set S of order at most k depends only on the isomorphism type of the subgraph induced by S. We classify the finite vertex-coloured k-ultrahomogeneous graphs and the finite vertex-coloured l-tuple regular graphs for k at least 4 and l at least 5, respectively. Our theorem in particular classifies finite vertex-coloured ultrahomogeneous graphs, where ultrahomogeneous mea…
We define a two-player combinatorial game in which players take alternate turns; each turn consists on deleting a vertex of a graph, together with all the edges containing such vertex. If any vertex became isolated by a player's move then it would also be deleted. A player wins the game when the other player has no moves available. We study this game under various viewpoints: by finding specific strategies for certain families of graphs, through using properties of a graph's automorphism group, by writing a program to look at Sprague-Grundy numbers, and by studying the game when played on rand…
Two vertices $a$ and $b$ in a graph $X$ are cospectral if the vertex-deleted subgraphs $X\setminus a$ and $X\setminus b$ have the same characteristic polynomial. In this paper we investigate a strengthening of this relation on vertices, that arises in investigations of continuous quantum walks. Suppose the vectors $e_a$ for $a$ in $V(X)$ are the standard basis for $\mathbb{R}^{V(X)}$. We say that $a$ and $b$ are strongly cospectral if, for each eigenspace $U$ of $A(X)$, the orthogonal projections of $e_a$ and $e_b$ are either equal or differ only in sign. We develop the basic theory of this co…
A graph admitting an automorphism with two orbits of the same length is called a bicirculant. Recently, Jajcay et al. initiated the investigation of the edge-transitive bicirculants with the properties that one of the subgraphs induced by the latter orbits is a cycle and the valence is at least $6$ (Electron. J. Combin., 2019). We show that the complement of the Petersen graph is the only such graph whose order is twice an odd number.
This paper deals with the Cayley graph $\Cay,$ where the generating set consists of all block transpositions. A motivation for the study of these particular Cayley graphs comes from current research in Bioinformatics. We prove that ${\rm{Aut}}(\Cay)$ is the product of the right translation group by $\textsf{N}\rtimes \textsf{D}_{n+1},$ where $\textsf{N}$ is the subgroup fixing $S_n$ element-wise and $\textsf{D}_{n+1}$ is a dihedral group of order $2(n+1)$. We conjecture that $\textsf{N}$ is trivial. We also prove that the subgraph $Γ$ with vertex-set $S_n$ is a $2(n-2)$-regular graph whose aut…
On the Expressibility of the Reconstructional Color Refinement
The Institute of Mathematical Sciences (HBNI)（数学科学研究所（HBNI））；Chennai Mathematical Institute（金奈数学研究所）；Institut für Informatik, Humboldt-Universität zu Berlin（柏林洪堡大学计算机科学研究所）；IAPMM
These notes concern aspects of various graphs whose vertex set is a group $G$ and whose edges reflect group structure in some way (so that they are invariant under the action of the automorphism group of $G$). The graphs I will discuss are the power graph, enhanced power graph, deep commuting graph, commuting graph, and non-generating graph, though I give a briefer discussion of the nilpotence and solvability graphs, and make some remarks on more general graphs. Aspects to be discussed include induced subgraphs, forbidden subgraphs, connectedness, and automorphism groups. We can also ask about…
We show that the only random orderings of finite graphs that are invariant under isomorphism and induced subgraph are the uniform random orderings. We show how this implies the unique ergodicity of the automorphism group of the random graph. We give similar theorems for other structures, including, for example, metric spaces. These give the first examples of uniquely ergodic groups, other than compact groups and extremely amenable groups, after Glasner and Weiss's example of the group of all permutations of the integers. We also contrast these results to those for certain special classes of gr…
A finite graph $\G$ is said to be {\em $(G,3)$-$($connected$)$ homogeneous} if every isomorphism between any two isomorphic (connected) subgraphs of order at most $3$ extends to an automorphism $g\in G$ of the graph, where $G$ is a group of automorphisms of the graph. In 1985, Cameron and Macpherson determined all finite $(G, 3)$-homogeneous graphs. In this paper, we develop a method for characterising $(G,3)$-connected homogeneous graphs. It is shown that for a finite $(G,3)$-connected homogeneous graph $\G=(V, E)$, either $G_v^{\G(v)}$ is $2$--transitive or $G_v^{\G(v)}$ is of rank $3$ and $…
A graph is said to be edge-transitive if its automorphism group acts transitively on its edges. It is known that edge-transitive graphs are either vertex-transitive or bipartite. In this paper we present a complete classification of all connected edge-transitive graphs on less than or equal to $20$ vertices. We then present a construction for an infinite family of edge-transitive bipartite graphs, and use this construction to show that there exists a non-trivial bipartite subgraph of $K_{m,n}$ that is connected and edge-transitive whenever $gcd(m,n)>2$. Additionally, we investigate necessary a…
We study the average probability that a discrete-time quantum walk finds a marked vertex on a graph. We first show that, for a regular graph, the spectrum of the transition matrix is determined by the weighted adjacency matrix of an augmented graph. We then consider the average search probability on a distance regular graph, and find a formula in terms of the adjacency matrix of its vertex-deleted subgraph. In particular, for any family of (1) complete graphs, or (2) strongly regular graphs, or (3) distance regular graphs of a fixed parameter $d$, varying valency $k$ and varying size $n$, such…
The projective norm graphs $\text{NG}(q,t)$ provide tight constructions for the Turán number of complete bipartite graphs $K_{t,s}$ with $s>(t-1)!$. In this paper we determine their automorphism group and explore their small subgraphs. To this end we give quite precise estimates on the number of solutions of certain equation systems involving norms over finite fields. The determination of the largest integer $s_t$, such that the projective norm graph $\text{NG}(q,t)$ contains $K_{t,s_t}$ for all large enough prime powers $q$ is an important open question with far-reaching general consequences.…
A relational structure is (connected-)homogeneous if every isomorphism between finite (connected) substructures extends to an automorphism of the structure. We investigate notions which generalise (connected-)homogeneity, where "isomorphism" may be replaced by "homomorphism" or "monomorphism" in the definition. Specifically, we study the classes of finite connected-homomorphism-homogeneous graphs, with the aim of producing classifications. The main result is a classification of the finite C-HH graphs, where a graph G is C-HH if every homomorphism from a finite connected induced subgraph of G i…
This paper focuses on vertices of the master corner polyhedra $P(G,g_0),$ the core of the group-theoretical approach to integer linear programming. We introduce two combinatorial operations that transform each vertex of $P(G,g_0)$ to adjacent ones. This implies that for any $P(G,g_0),$ there exists a subset of basic vertices, we call them support vertices, from which all others can be built. The class of support vertices is proved to be invariant under the automorphism group of $G,$ so this basis can be further reduced to a subset of pairwise non-equivalent support vertices. Among other result…
We use crystals of tableaux and descent compositions to understand the decomposition of Schur functions $s_λ$ into Gessel's fundamental quasisymmetric functions $F_α$. The connected crystal of tableaux $B(λ)$, associated to $s_λ$, is shown to be partitionned into a disjoint union of connected induced subgraphs $B(T_α)$ corresponding to the $F_α$'s. We show that these subgraphs, which we call quasicrystals, are isomorphic (as graphs) to specific crystals of tableaux. This allows us to give a formula for the number of tableaux of shape $λ$ and maximal entry $n$. We also use this setting to give …
We study locally closed transformation monoids which contain the automorphism group of the random graph. We show that such a transformation monoid is locally generated by the permutations in the monoid, or contains a constant operation, or contains an operation that maps the random graph injectively to an induced subgraph which is a clique or an independent set. As a corollary, our techniques yield a new proof of Simon Thomas' classification of the five closed supergroups of the automorphism group of the random graph; our proof uses different Ramsey-theoretic tools than the one given by Thomas…
Perturbation results for distance-edge-monitoring numbers
Qinghai Normal University（青海师范大学）；Université de Bordeaux（波尔多大学）；Bordeaux INP（波尔多理工学院）；University of Shanghai for Science and Technology（上海理工大学）；School of Mathematics and Statistics（数学与统计学院）
