Sebastian Jeon, Tanya Khovanova
An intuitive property of a random graph is that its subgraphs should also appear randomly distributed. We consider graphs whose subgraph densities exactly match their expected values. We call graphs with this property for all subgraphs with $k$ vertices to be $k$-symmetric. We discuss some properties and examples of such graphs. We construct 3-symmetric graphs and provide some statistics.
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We construct infinitely many signed graphs having symmetric spectrum, by using the NEPS and rooted product of signed graphs. We also present a method for constructing large cospectral signed graphs. Although the obtained family contains only a minority of signed graphs, it strengthen the belief that the signed graphs with symmetric spectrum are deeper than bipartite graphs, i.e the unsigned graphs with symmetric spectrum.
Consider a randomly oriented graph $G=(V,E)$ and let $a$, $s$ and $b$ be three distinct vertices in $V$. We study the correlation between the events $\{a\to s\}$ and $\{s\to b\}$. We show that, when $G$ is the complete graph $K_n$, the correlation is negative for $n=3$, zero for $n=4$, and that, counter-intuitively, it is positive for $n\ge 5$. We also show that the correlation is always negative when $G$ is a cycle, $C_n$, and negative or zero when $G$ is a tree (or a forest).
A signed graph is said to be sign-symmetric if it is switching isomorphic to its negation. Bipartite signed graphs are trivially sign-symmetric. We give new constructions of non-bipartite sign-symmetric signed graphs. Sign-symmetric signed graphs have a symmetric spectrum but not the other way around. We present constructions of signed graphs with symmetric spectra which are not sign-symmetric. This, in particular answers a problem posed by Belardo, Cioabă, Koolen, and Wang (2018).
We show that if the two parts of a finite bipartite graph have the same degree sequence, then there is a bipartite graph, with the same degree sequences, which is symmetric, in that it has an involutive graph automorphism that interchanges its two parts. To prove this, we study the relationship between symmetric bipartite graphs and graphs with loops.
The signed graphs with symmetric spectra
本文研究具有对称谱的符号图的刻画问题，给出了谱对称符号图的必要和（或）充分条件，并提出了构造此类符号图的方法，生成了无限族符号图。
A plabic graph is a planar bicolored graph embedded in a disk, which satisfies some combinatorial conditions. Postnikov's boundary measurement map takes the space of positive edge weights of a plabic graph $G$ to a positroid cell in some totally nonnegative Grassmannian. In this note, we investigate plabic graphs which are symmetric about a line of reflection, up to reversing the colors of vertices. These symmetric plabic graphs arise naturally in the study of total positivity for the Lagrangian Grassmannian. We characterize various combinatorial objects associated with symmetric plabic graphs…
This paper describes how many known graph polynomials arise from the coefficients of chromatic symmetric function expansions in different bases, and studies a new polynomial arising by expanding over a basis given by chromatic symmetric functions of trees.
Richard P. Stanley defined the chromatic symmetric function of a simple graph and has conjectured that every tree is determined by its chromatic symmetric function. Recently, Takahiro Hasebe and the author proved that the order quasisymmetric functions, which are analogs of the chromatic symmetric functions, distinguish rooted trees. In this paper, using a similar method, we prove that the chromatic symmetric functions distinguish trivially perfect graphs. Moreover, we also prove that claw-free cographs, that is, $ \{K_{1,3},P_{4}\} $-free graphs belong to a known class of $ e $-positive graph…
In this paper, we derive an upper bound for higher order eigenvalues of the normalized Laplace operator associated with a symmetric finite graph in terms of lower order eigenvalues.
A connected symmetric graph of prime valency is {\em basic} if its automorphism group contains no nontrivial normal subgroup having more than two orbits. Let $p$ be a prime and $n$ a positive integer. In this paper, we investigate properties of connected pentavalent symmetric graphs of order $2p^n$, and it is shown that a connected pentavalent symmetric graph of order $2p^n$ is basic if and only if it is either a graph of order $6$, $16$, $250$, or a graph of three infinite families of Cayley graphs on generalized dihedral groups -- one family has order $2p$ with $p=5$ or $5 \mid (p-1)$, one f…
A graph $G$ is called \emph{symmetric with respect to a functional $F_G(P)$} defined on the set of all the probability distributions on its vertex set if the distribution $P^*$ maximizing $F_G(P)$ is uniform on $V(G)$. Using the combinatorial definition of the entropy of a graph in terms of its vertex packing polytope and the relationship between the graph entropy and fractional chromatic number, we prove that vertex transitive graphs are symmetric with respect to graph entropy. As the main result of this paper, we prove that a perfect graph is symmetric with respect to graph entropy if and on…
We present some graphical characterizations of positive definite symmetric quasi-Cartan matrices of Dynkin type $\mathbb{A}_{n}$ and $\mathbb{D}_{n}$. Our proofs are constructive, purely graph theoretical, and almost self-contained in the sense that they rely on the classical inflations method only.
A finite graph $Γ$ is called $G$-symmetric if $G$ is a group of automorphisms of $Γ$ which is transitive on the set of ordered pairs of adjacent vertices of $Γ$. We study a family of symmetric graphs, called the unitary graphs, whose vertices are flags of the Hermitian unital and whose adjacency relations are determined by certain elements of the underlying finite fields. Such graphs admit the unitary groups as groups of automorphisms, and they play a significant role in the classification of a family of symmetric graphs with complete quotients such that an associated incidence structure is a …
Two celebrated conjectures in chromatic symmetric function theory concern the positivity chromatics symmetric functions of claw-free graphs. Here we extend the claw-free idea to general graphs and consider the e-positivity question for H-free graphs where H = {claw, F} and H={claw, F, co-F}, where F is a four-vertex graph. We settle the question for all cases except H={claw, co-diamond}, and we provide some partial results in that case.
Let $Γ$ be a graph and let $G$ be a group of automorphisms of $Γ$. The graph $Γ$ is called $G$-normal if $G$ is normal in the automorphism group of $Γ$. Let $T$ be a finite non-abelian simple group and let $G = T^l$ with $l\geq 1$. In this paper we prove that if every connected pentavalent symmetric $T$-vertex-transitive graph is $T$-normal, then every connected pentavalent symmetric $G$-vertex-transitive graph is $G$-normal. This result, among others, implies that every connected pentavalent symmetric $G$-vertex-transitive graph is $G$-normal except $T$ is one of $57$ simple groups. Furthermo…
Aldous [1] asked whether every graph in which the distribution of the return time of random is independent of the starting vertex must be transitive. We remark that this question can be reduced into a purely graph-theoretic one that had already been answered Godsil & McKay [6] and ask some questions motivated by this.
Constructing reflection-symmetric flexible realisations of graphs
本文提出了一种基于边着色的条件，用于构造具有反射对称性的图的柔性实现，并研究了由三角形和平行四边形组成的特定反射对称性实现类别。
Symmetric tensor powers of graphs
本文引入图的对称张量幂概念，研究其基本性质与有趣现象，并提出多个开放问题以推动后续研究。
Stanley [9] introduced the chromatic symmetric function ${\bf X}_G$ associated to a simple graph $G$ as a generalization of the chromatic polynomial of $G$. In this paper we present a novel technique to write ${\bf X}_G$ as a linear combination of chromatic symmetric functions of smaller graphs. We use this technique to give a sufficient condition for two graphs to have the same chromatic symmetric function. We then construct an infinite family of pairs of unicyclic graphs with the same chromatic symmetric function, answering the question posed by Martin, Morin, and Wagner [7] of whether such …
A graph $\G$ is {\em symmetric} or {\em arc-transitive} if its automorphism group $\Aut(\G)$ is transitive on the arc set of the graph, and $\G$ is {\em basic} if $\Aut(\G)$ has no non-trivial normal subgroup $N$ such that the quotient graph $\G_N$ has the same valency with $\G$. In this paper, we classify symmetric basic graphs of order $2qp^n$ and valency 5, where $q<p$ are two primes and $n$ is a positive integer. It is shown that such a graph is isomorphic to a family of Cayley graphs on dihedral groups of order $2q$ with $5\di (q-1)$, the complete graph $K_6$ of order $6$, the complete bi…
It was proved in [Y.-Q. Feng, C. H. Li and J.-X. Zhou, Symmetric cubic graphs with solvable automorphism groups, {\em European J. Combin.} {\bf 45} (2015), 1-11] that a cubic symmetric graph with a solvable automorphism group is either a Cayley graph or a $2$-regular graph of type $2^2$, that is, a graph with no automorphism of order $2$ interchanging two adjacent vertices. In this paper an infinite family of non-Cayley cubic $2$-regular graphs of type $2^2$ with a solvable automorphism group is constructed. The smallest graph in this family has order 6174.
This paper has two main parts. First, we consider the Tutte symmetric function $XB$, a generalization of the chromatic symmetric function. We introduce a vertex-weighted version of $XB$ and show that this function admits a deletion-contraction relation. We also demonstrate that the vertex-weighted $XB$ admits spanning-tree and spanning-forest expansions generalizing those of the Tutte polynomial by connecting $XB$ to other graph functions. Second, we give several methods for constructing nonisomorphic graphs with equal chromatic and Tutte symmetric functions, and use them to provide specific e…
For some positive integer $k$, if the finite cyclic group $\mathbb{Z}_k$ can act freely on a graph $G$, then we say that $G$ is $k$-symmetric. In 1985, Faria showed that the multiplicity of Laplacian eigenvalue 1 is greater than or equal to the difference between the number of pendant vertices and the number of quasi-pendant vertices. But if a graph has a pendant vertex, then it is at most 1-connected. In this paper, we investigate a class of 2-connected $k$-symmetric graphs with a Laplacian eigenvalue 1. We also identify a class of $k$-symmetric graphs in which all Laplacian eigenvalues are i…
The Tutte symmetric matrix of a graph
本文提出基于矩阵的图特对称函数公式，通过构造无限矩阵M_G，展示图的拼接对应矩阵乘法，头尾拼接对应迹运算，图的反转对应转置。
Let $F_G(P)$ be a functional defined on the set of all the probability distributions on the vertex set of a graph $G$. We say that $G$ is \emph{symmetric with respect to $F_G(P)$} if the uniform distribution on $V(G)$ maximizes $F_G(P)$. Using the combinatorial definition of the entropy of a graph in terms of its vertex packing polytope and the relationship between the graph entropy and fractional chromatic number, we characterize all graphs which are symmetric with respect to graph entropy. We show that a graph is symmetric with respect to graph entropy if and only if its vertex set can be un…
In this paper we describe all edge-colored graphs that are fully symmetric with respect to colors and transitive on every set of edges of the same color. They correspond to fully symmetric homogeneous factorizations of complete graphs. Our description completes the work done in our previous paper, where we have shown, in particular, that there are no such graphs with more than 5 colors. Using some recent results, with a help of computer, we settle all the cases that was left open in the previous paper.
Stanley introduced the chromatic symmetric function of a simple graph, which is a generalization of a chromatic polynomial. This is expressed in terms of the integer points of the complements of the corresponding graphic arrangement. Stanley proved a combinatorial reciprocity theorem for chromatic functions. This is considered as an Ehrhart-type reciprocity theorem for the graphic arrangement. We introduce the chromatic signed-symmetric function of a signed graph, an analogue of the chromatic symmetric function, by the integer points of the complements of the corresponding signed-graphic arran…
If V is the vertex sequence of a symmetric 2t-cycle in the hypercube graph with the vertices {1,-1}^t, then for any vertex T of the graph there exists a unique inclusion-minimal subset of V such that T is the sum of its elements. We present a simple combinatorial statistic on decompositions of vertices of the hypercube graphs with respect to symmetric cycles and describe their basic metric properties.
Finite connected cubic symmetric graphs of girth 6 have been classified by K. Kutnar and D. Marušič, in particular, each of these graphs has an abelian automorphism group with two orbits on the vertex set. In this paper all cubic symmetric graphs with the latter property are determined. In particular, with the exception of the graphs K_4, K_{3,3}, Q_3, GP(5,2), GP(10,2), F40 and GP(24,5), all the obtained graphs are of girth 6.
We investigate properties of signed graphs that have few distinct eigenvalues together with a symmetric spectrum. Our main contribution is to determine all signed $(0,2)$-graphs with vertex degree at most $6$ that have precisely two distinct eigenvalues $\pm λ$. Next, we consider to what extent induced subgraphs of signed graph with two distinct eigenvalues $\pm λ$ are determined by their spectra. Lastly, we classify signed $(0,2)$-graphs that have a symmetric spectrum with three distinct eigenvalues and give a partial classification for those with four distinct eigenvalues.
Bispanning graphs are undirected graphs with an edge set that can be decomposed into two disjoint spanning trees. The operation of symmetrically swapping two edges between the trees, such that the result is a different pair of disjoint spanning trees, is called an edge exchange or a symmetric base exchange. The graph of symmetric base exchanges of a bispanning graph contains a vertex for every valid pair of disjoint spanning trees, and edges between them to represent all possible edge exchanges. We are interested in a restriction of these graphs to only unique symmetric base exchanges, which a…
Chromatic symmetric functions of conjoined graphs
本文研究了联图的色对称函数，通过组合方法得到了几种图类的色对称函数展开式，并提出了帽链的 e 正猜想。
In this expository paper we describe a group theoretic characterization of arc-transitive graphs and their quotients. When passing from a symmetric graph to its quotient, much information is lost, but some of this information may be recovered from a certain combinatorial design on the blocks, as well as a bipartite graph between the blocks. We address the "extention problem" which asks, when is this additional information sufficient to recover the original graph?
We present explicit descriptions of the decompositions of vertices of a hypercube graph with respect to its distinguished symmetric cycle.
We present statistics on the decompositions (with respect to a distinguished symmetric 2t-cycle) of vertices of the hypercube graph, whose negative parts are covered by two subsets of the ground set {1,...,t} of the corresponding oriented matroid.
A graph is said to be symmetric if its automorphism group is transitive on its arcs. Guo et al. (Electronic J. Combin. 18, \#P233, 2011) and Pan et al. (Electronic J. Combin. 20, \#P36, 2013) determined all pentavalent symmetric graphs of order $4pq$. In this paper, we shall generalize this result by determining all connected pentavalent symmetric graphs of order four times an odd square-free integer. It is shown in this paper that, for each of such graphs $\itΓ$, either the full automorphism group ${\sf Aut}\itΓ$ is isomorphic to ${\sf PSL}(2,p)$, ${\sf PGL}(2,p)$, ${\sf PSL}(2,p){\times}\mat…
For a graph G, M(G) denotes the maximum multiplicity occurring of an eigenvalue of a symmetric matrix whose zero-nonzero pattern is given by edges of G. We introduce two combinatorial graph parameters T^-(G) and T^+(G) that give a lower and an upper bound for M(G) respectively, and we show that these bounds are sharp.
In 1995, Richard Stanley introduced the chromatic symmetric function $X_G$ of a graph $G$ and proved that, when written in terms of the elementary symmetric functions, it reveals the number of acyclic orientations of $G$ with a given number of sinks. In this paper, we generalize this result to signed graphs, that is, to graphs whose edges are labeled with $+$ or $-$ and whose colorings and orientations can interact with their signs. Additionally, we introduce a non-homogeneous basis which detects the number of sinks and which not only gives a Stanley-type result for signed graphs but gives an …
Number of facets of symmetric edge polytopes arising from join graphs
本文研究了连通图对称边多面体面数的上下界猜想，证明了该猜想对任意两图的联图成立，从而为 Nill 关于反身多面体面数上界的猜想提供了部分回答。
Spectrally symmetric orientations of graphs
本文研究了非二分图的谱对称定向图，探讨了其密度极值问题及构造方法。
A regular cover of a connected graph is called {\em cyclic} or {\em dihedral} if its transformation group is cyclic or dihedral respectively, and {\em arc-transitive} (or {\em symmetric}) if the fibre-preserving automorphism subgroup acts arc-transitively on the regular cover. In this paper, we give a classification of arc-transitive cyclic and dihedral covers of a connected pentavalent symmetric graph of order twice a prime. All those covers are explicitly constructed as Cayley graphs on some groups, and their full automorphism groups are determined.
For a graph $G$ with vertex set $V_{G}$ and edge set $E_{G}$, the symmetric division deg index is defined as $SDD(G)=\sum\limits_{uv\in E_{G}}(\frac{d_{u}}{d_{v}}+\frac{d_{v}}{d_{u}})$, where $d_{u}$ denotes the degree of vertex $u$ in $G$. In 2018, Furtula et al. confirmed the quality of SDD index exceeds that of some more popular VDB indices, in particular that of the GA index. They shown a close connection between the SDD index and the earlier well-established GA index. Thus it is meaningful and important to consider the chemical and mathematical properties of the SDD index. In this paper, …
Let $\mathfrak g$ be a Borcherds algebra with the associated graph $G$. We prove that the chromatic symmetric function of $G$ can be recovered from the Weyl denominator identity of $\mathfrak g$ and this gives a Lie theoretic proof of Stanley's expression for chromatic symmetric function in terms of power sum symmetric function. Also, this gives an expression for chromatic symmetric function of $G$ in terms of root multiplicities of $\lie g$. The absolute value of the linear coefficient of the chromatic polynomial of $G$ is known as the chromatic discriminant of $G$. As an application of our m…
We present statistics on the decompositions (with respect to a distinguished symmetric 2t-cycle) of vertices of the hypercube graph, whose negative parts are regarded as disjoint unions of two subsets of the ground set {1,...,t} of the corresponding oriented matroid.
How to find all extremal graphs using symmetric subgraphs
本文基于对称子图概念，研究如何通过构造对称子图来确定特定图族的极值图，并取得新的极值结果。
In this note we consider $k$-regular multigraphs, where the possible edge multiplicities are controlled. These structures are considered in a question recently posed by Brendan McKay. We express the generating functions using the scalar product of symmetric functions, and consequently give conditions on when the classes are D-finite. We appeal to symmetric species results of Mendez to write the expressions in a systematic way.
The present work is devoted to characterize the family of symmetric undirected Cayley graphs over finite Abelian groups for degrees 4 and 6.
The complete symmetric directed graph of order $v$, denoted $K_{v}^*$, is the directed graph on $v$ vertices that contains both arcs $(x,y)$ and $(y,x)$ for each pair of distinct vertices $x$ and $y$. For a given directed graph, $D$, the set of all $v$ for which $K_{v}^*$ admits a $D$-decomposition is called the spectrum of $D$. There are 10 non-isomorphic orientations of a $7$-cycle (heptagon). In this paper, we completely settled the spectrum problem for each of the oriented heptagons.
The main result of this paper is the introduction of marked graphs and the marked graph polynomials ($M$-polynomial) associated with them. These polynomials can be defined via a deletion-contraction operation. These polynomials are a generalization of the $W$-polynomial introduced by Noble and Welsh and a specialization of the $\mathbf{V}$-polynomial introduced by Ellis-Monaghan and Moffatt. In addition, we describe an important specialization of the $M$-polynomial which we call the $D$-polynomial. Furthermore, we give an efficient algorithm for computing the chromatic symmetric function of a …
Symmetric Grothendieck polynomials are analogues of Schur polynomials in the K-theory of Grassmannians. We build dual families of symmetric Grothendieck polynomials using Schur operators. With this approach we prove skew Cauchy identity and then derive various applications: skew Pieri rules, dual filtrations of Young's lattice, generating series and enumerative identities. We also give a new explanation of the finite expansion property for products of Grothendieck polynomials.
Structure of quasi-crystal graphs and applications to the combinatorics of quasi-symmetric functions
本文通过引入拟阵列结构，研究了拟晶体图的组合性质，解释了其分量同构，并分别肯定和否定了Maas-Gariépy关于拟对称函数与Schur函数相互作用及分量排列的两个猜想。
Rotationally symmetric plabic graphs and the Lagrangian Grassmannian
本文引入了旋转对称的plabic图来表示总非负拉格朗日格拉斯曼流形，并发展了处理非简化的plabic图的新技术。
A graph $Γ$ is basic if Aut$Γ$ has no normal subgroup $N\ne1$ such that $Γ$ is a normal cover of the normal quotient graph $Γ_N$. In this paper, we completely determine the basic normal quotient graphs of all connected 7-valent symmetric graphs of order $2pq^n$ with $p < q$ odd primes, which consist of an infinite family of dihedrants of order $2p$ with $p\equiv1$(mod 7), and 6 specific graphs with order at most 310. As a consequence, it shows that, for any given positive integer n, there are only finitely many connected 2-arc-transitive 7-valent graphs of order $2pq^n$ with $7\ne p<q$ primes,…
We characterise the quotient surface graphs arising from symmetric contact systems of line segments in the plane and also from symmetric pointed pseudotriangulations in the case where the group of symmetries is generated by a translation or a rotation of finite order. These results generalise well known results of Thomassen, in the case of line segments, and of Streinu and Haas et al., in the case of pseudotriangulations. Our main tool is a new inductive characterisation of the appropriate classes of surface graphs. We also discuss some consequences of our results in the area of geometric rigi…
Local fractional metric dimension of rotationally symmetric planar graphs arisen from planar chorded cycles
本文通过平面弦环的边并合构造新的旋转对称平面图族，确定其局部分数度量维数，并利用渐近行为证明存在常数或有界局部分数维数的图族。
Edge pancyclic Cayley graphs on symmetric group
本文证明n≥4时对称群上的错位排列图Γₙ是边泛圈的，并将结果推广到对称群上两类更广泛的凯莱图，拓展了凯莱图的结构性质研究。
We study the equivalence between bipartiteness and symmetry of spectra of mixed graphs, for $θ$-Hermitian adjacency matrices defined by an angle $θ\in (0, π]$. We show that this equivalence holds when, for example, an angle $θ$ is an algebraic number, while it breaks down for any angle $θ\in \mathbb{Q}π$. Furthermore, we construct a family of non-bipartite mixed graphs having the symmetric spectra for given $θ\in \mathbb{Q}π$.
Hypergraph expanders are hypergraphs with surprising, non-intuitive expansion properties. In a recent paper, the first author gave a simple construction, which can be randomized, of $3$-uniform hypergraph expanders with polylogarithmic degree. We generalize this construction, giving a simple construction of $r$-uniform hypergraph expanders for all $r \geq 3$.
Prime-valent Symmetric Cayley Graphs of Characteristically Simple Groups
本文刻画了特征单群$T^k$（$k\geq2$）的连通本原度弧传递Cayley图的结构，并证明三次情形下所有连通对称Cayley图均为正规图。
In the mid-1990s, Stanley and Stembridge conjectured that the chromatic symmetric functions of claw-free co-comparability (also called incomparability) graphs were e-positive. The quest for the proof of this conjecture has led to an examination of other, related graph classes. In 2013 Guay-Paquet proved that if unit interval graphs are e-positive, that implies claw-free incomparability graphs are as well. Inspired by this approach, we consider a related case and prove that unit interval graphs whose complement is also a unit interval graph are e-positive. We introduce the concept of strongly $…
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