Teodor Banica, Julien Bichon, Gaetan Chenevier
We consider circulant graphs having $p$ vertices, with $p$ prime. To any such graph we associate a certain number $k$, that we call type of the graph. We prove that for $p>>k$ the graph has no quantum symmetry, in the sense that the quantum automorphism group reduces to the classical automorphism group.
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The set of semialgebraic graphs having countable list-chromatic numbers is characterized. Some other related sets of graphs having countable list-chromatic numbers also are.
In this paper, we study the non-bipartite distance-regular graphs with valency k and having a smallest eigenvalue at most -k/2.
In this paper we show that for each $n$, there exists a simple graph whose reliability polynomial has at least $n$ inflection points.
For a given positive integer t we consider graphs having maximal independent sets of precisely t distinct cardinalities and restrict our attention to those that have no vertices of degree one. In the situation when t is four or larger and the length of the shortest cycle is at least 6t-6, we completely characterize such graphs.
A mixed graph is obtained by orienting some edges of a simple graph. The positive inertia index of a mixed graph is defined as the number of positive eigenvalues of its Hermitian adjacency matrix, including multiplicities. This matrix was introduced by Liu and Li, independently by Guo and Mohar, in the study of graph energy. Recently, Yuan et al. characterized the mixed graphs with exactly one positive eigenvalue. In this paper, we study the positive inertia indices of mixed graphs and characterize the mixed graphs with cut vertices having positive inertia index 2.
We prove that every simple connected graph with no $K_5$ minor admits a proper 4-coloring such that the neighborhood of each vertex $v$ having more than one neighbor is not monochromatic, unless the graph is isomorphic to the cycle of length 5. This generalizes the result by S.-J. Kim, S. J. Lee, and W.-J. Park on planar graphs.
Let $H$ and $G$ be graphs on $n$ vertices, where $n$ is sufficiently large. We prove that if $H$ has Ore-degree at most 5 and $G$ has minimum degree at least $2n/3$ then $H\subset G.$
The eccentricity (anti-adjacency) matrix $\varepsilon(G)$ of a graph $G$ is obtained from the distance matrix by retaining the eccentricities in each row and each column. This matrix is first defined in 2018 by Wang et al. \cite{1}. In this paper we have characterized the graphs which have at most one (hence exactly) positive eigenvalue of $\varepsilon(G)$.
The parameter $σ(G)$ of a graph $G$ stands for the number of Laplacian eigenvalues greater than or equal to the average degree of $G$. In this work, we address the problem of characterizing those graphs $G$ having $σ(G)=1$. Our conjecture is that these graphs are stars plus a (possible empty) set of isolated vertices. We establish a link between $σ(G)$ and the number of anticomponents of $G$. As a by-product, we present some results which support the conjecture, by restricting our analysis to some classes of graphs.
The vertex $k$-partiteness $v_k(G)$ of graph $G$ is defined as the fewest number of vertices whose deletion from $G$ yields a $k$-partite graph. In this paper, we introduce two concepts: monotonic decreasing topological index and monotonic increasing topological index, and characterize the extremal graphs having the minimum Wiener index, the maximum Harry index, the maximum reciprocal degree distance, the minimum eccentricity distance sum, the minimum adjacent eccentric distance sum index, the maximum connective eccentricity index, the maximum Zagreb indices among graphs with a fixed number $n…
In this paper, we give some sufficient conditions for graphs with an edge perturbation between twin vertices to have Laplacian perfect pair state transfer as well as Laplacian pretty good pair state transfer. By those sufficient conditions, we also construct many new graphs having Laplacian perfect pair state transfer as well as Laplacian pretty good pair state transfer.
In this work, the classical Nelson -- Hadwiger problem is studied which lies on the edge of combinatorial geometry and graph theory. It concerns colorings of distance graphs in $ {\mathbb R}^n $, i.e., graphs such that their vertices are vectors and their edges are pairs of vectors at a distance from a given set of postive numbers apart. A series of new lower bounds are obtained for the chromatic numbers of such graphs with different restrictions on the clique numbers and the girths.
Let $G$ be a graph with adjacency matrix $A$. The transition matrix corresponding to $G$ is defined by $H(t):=\exp{\left(itA\right)}$, $t\in\Rl$. The graph $G$ is said to have perfect state transfer (PST) from a vertex $u$ to another vertex $v$, if there exist $τ\in\Rl$ such that the $uv$-th entry of $H(τ)$ has unit modulus. The graph $G$ is said to be periodic at $τ\in\Rl$ if there exist $γ\in\Cl$ with $|γ|=1$ such that $H(τ)=γI$, where $I$ is the identity matrix. A $\mathit{gcd}$-graph is a Cayley graph over a finite abelian group defined by greatest common divisors. In this paper, we constr…
Motivated by a result of [17], we determine necessary and sufficient conditions on $F\/$ with $|E(F)| \leq n-1\/$ for which $K_n - F\/$ admits a $g$-angulation. For $|E(F)| \geq n\/$, we investigate the possibility of placing $F\/$ in $K_n\/$ such that $K_n -F\/$ admits a $g$-angulation for certain families of graphs $F\/$.
Planar graphs having no cycle of length $4$, $6$ or $8$ are DP-3-colorable
本文证实了Dvořák和Postle的猜想，通过证明更强的结果：没有长度为4、6或8的圈的平面图是DP-3-可着色的。
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.
Let $\mathcal{B}(n,q)$ be the class of block graphs on $n$ vertices having all their blocks of the same size. We prove that if $G\in \mathcal{B}(n,q)$ has at most three pairwise adjacent cut vertices then the minimum spectral radius $ρ(G)$ is attained at a unique graph. In addition, we present a lower bound for $ρ(G)$ when $G\in \mathcal{B}(n,q)$.
Revisiting Extremal Graphs Having No Stable Cutsets
本文填补了Le和Pfender关于边数为2n-3且无稳定割集的极图递归构造证明中的漏洞，完善了对Caro猜想相关极图结构的论证。
Let $Γ$ be a distance-regular graph with diameter $d$ and Kneser graph $K=Γ_d$, the distance-$d$ graph of $Γ$. We say that $Γ$ is partially antipodal when $K$ has fewer distinct eigenvalues than $Γ$. In particular, this is the case of antipodal distance-regular graphs ($K$ with only two distinct eigenvalues), and the so-called half-antipodal distance-regular graphs ($K$ with only one negative eigenvalue). We provide a characterization of partially antipodal distance-regular graphs (among regular graphs with $d$ distinct eigenvalues) in terms of the spectrum and the mean number of vertices at m…
Let $a\leq b$ be two positive integers. We say that a graph $G$ has all $[a,b]$-factors if it has an $h$-factor for every function $h: V(G)\rightarrow \mathbb{Z}^+$ such that $a\le h(v) \le b$ for all $v\in V(G)$ and $\sum_{v\in V(G)}h(v)\equiv 0\pmod 2$, and has all fractional $[a,b]$-factors if it has a fractional $p$-factor for every $p: V(G) \rightarrow \mathbb{Z}^+$ such that $a\le p(v)\le b$ for all $v\in V(G)$. In this paper, we provide tight spectral radius conditions for graphs having all $[a,b]$-factors ($3\leq a<b$) and all fractional $[a,b]$-factors ($1\leq a<b$), respectively.
A graph $G$ has the \emph{strong parity property} if for every subset $X\subseteq V$ with $|X|$ even, $G$ has a spanning subgraph $F$ with minimum degree at least one such that $d_F(v)\equiv 1\pmod 2$ for all $v\in X$, $d_F(y)\equiv 0\pmod 2$ for all $y\in V(G)-X$. Bujtás, Jendrol and Tuza (On specific factors in graphs, \emph{Graphs and Combin.}, 36 (2020), 1391-1399.) introduced the concept and conjectured that every 2-edge-connected graph with minimum degree at least three has the strong parity property. In this paper, we give a characterization for graphs to have the strong parity property…
We completely describe all integer symmetric matrices that have all their eigenvalues in the interval [-2,2]. Along the way we classify all signed graphs, and then all charged signed graphs, having all their eigenvalues in this same interval. We then classify subsets of the above for which the integer symmetric matrices, signed graphs and charged signed graphs have all their eigenvalues in the open interval (-2,2).
In this paper, we consider the problem on the existence of perfect state transfer(PST for short) on semi-Cayley graphs over abelian groups (which are not necessarily regular), i.e on the graphs having semiregular and abelian subgroups of automorphisms with two orbits of equal size. We stablish a characterization of semi-Cayley graphs over abelian groups having PST. As a result, we give a characterization of Cayley graphs over groups with an abelian subgroup of index 2 having PST, which improves the earlier results on Cayley graphs over abelian groups, dihedral groups and dicyclic group and det…
Given a finite group G, the bipartite divisor graph for its conjugacy class sizes is the bipartite graph with bipartition consisting of the set of conjugacy class sizes of G-Z (where Z denotes the centre of G) and the set of prime numbers that divide these conjugacy class sizes, and with {p,n} being an edge if gcd(p,n)\neq 1. In this paper we construct infinitely many groups whose bipartite divisor graph for their conjugacy class sizes is the complete bipartite graph K_{2,5}, giving a solution to a question of Taeri.
A set $P$ of vertices in a graph $G$ is an open packing if no two distinct vertices in $P$ have a common neighbor. Among all maximal open packings in $G$, the smallest cardinality is denoted $ρ^{\rm o}_L(G)$ and the largest cardinality is $ρ^{\rm o}(G)$. There exist graphs for which these two invariants are arbitrarily far apart. In this paper we begin the investigation of the class of graphs that have one size of maximal open packings. By presenting a method of constructing such graphs we show that every graph is the induced subgraph of a graph in this class. The main result of the paper is a…
In 1975, Plesník characterized all triangle-free planar graphs as having a diameter $2$. We characterize all triangle-free projective-planar graphs having a diameter $2$ and discuss some applications. In particular, the main result is applied to calculate the analogue of clique numbers for graphs, namely, colored mixed graphs, having different types of arcs and edges.
A spectral condition for a graph having a strong parity factor
该研究针对图的强奇偶因子存在性问题，提出了一种谱半径条件，可用于判定连通图是否包含强奇偶因子，为图论中相关奇偶因子问题的研究提供了谱方法视角的结论。
Let $G$ be a connected simple graph of order $n$. Let $ρ_1(G)\geq ρ_2(G)\geq \cdots \geq ρ_{n-1}(G)> ρ_n(G)=0$ be the eigenvalues of the normalized Laplacian matrix $\mathcal{L}(G)$ of $G$. Denote by $m(ρ_i)$ the multiplicity of the normalized Laplacian eigenvalue $ρ_i$. Let $ν(G)$ be the independence number of $G$. In this paper, we give a full characterization of graphs with some normalized Laplacian eigenvalue of multiplicity $n-3$, which answers a remaining problem in [S. Sun, K.C. Das, On the multiplicities of normalized Laplacian eigenvalues of graphs, Linear Algebra Appl. 609 (2021) 365…
In this paper we define some new labellings for trees, called the in-improper and out-improper odd-graceful labellings such that some trees labelled with the new labellings can induce graceful graphs having at least a cycle. We, next, apply the new labellings to construct large scale of graphs having improper graceful/odd-graceful labellings or having graceful/odd-graceful labellings.
Let $G$ be a graph with order $n$ and adjacency matrix $\mathbf{A}(G)$. The adjacency polynomial of $G$ is defined as $ϕ(G;λ) =det(λ\mathbf{I}-\mathbf{A}(G))=\sum_{i=0}^n\mathbf{a_i}(G)λ^{n-i}$. Hereafter, $\mathbf{a}_i(G)$ is called the $i$-th adjacency coefficient of $G$. Denote by $\mathfrak{G}_{n,m}$ the set of all connected graphs having $n$ vertices and $m$ edges. A graph $G$ is said $4$-Sachs minimal if $$\mathbf{a}_4(G)=min\{\mathbf{a}_4(H)|H\in \mathfrak{G}_{n,m}\}.$$ The value $min\{\mathbf{a}_4(H)|H\in \mathfrak{G}_{n,m}\}$ is called the minimal $4$-Sachs number in $\mathfrak{G}_{n,…
Characterizing tricyclic graphs with pendant vertices having largest $A_α$-spectral radius
该研究刻画了含k个悬挂点的n阶三圈图中α∈[1/2,1)时具有最大A_α-谱半径的唯一图，并推导了保证含k个悬挂点的图无三圈结构的充分谱条件。
Let $G$ be a simple graph with order $n$ and adjacency matrix $\mathbf{A}(G)$. Let $ϕ(G; λ)=\det(λI-\mathbf{A}(G))=\sum_{i=0}^n\mathbf{a}_i(G)λ^{n-i}$ be the characteristic polynomial of $G$, where $\mathbf{a}_i(G)$ is called the $i$-th adjacency coefficient of $G$. Denote by $\mathfrak{B}_{n,m}$ the set of all connected graphs having $n$ vertices and $m$ edges. A bipartite graph $G$ is referred as bipartite optimal if $$\mathbf{a}_4(G)=min\{\mathbf{a}_4(H)|H\in \mathfrak{B}_{n,m}\}.$$ The value $min\{\mathbf{a}_4(H)|H\in \mathfrak{B}_{n,m}\}$ is called the minimal $4$-Sachs number in $\mathfr…
A formula for computing the metric dimension of a simple graph, having no singleton twin, is given. A sufficient condition for a simple graph to have the exchange property, for resolving sets, is found. Some families of power graphs of finite groups, having this exchange property, are identified. The metric dimension of the power graph of a dihedral group is also computed.
A half-square of a bipartite graph $B=(X,Y,E_B)$ has one color class of $B$ as vertex set, say $X$; two vertices are adjacent whenever they have a common neighbor in $Y$. If $G=(V,E_G)$ is the half-square of a planar bipartite graph $B=(V,W,E_B)$, then $G$ is called a map graph, and $B$ is a witness of $G$. Map graphs generalize planar graphs, and have been introduced and investigated by Chen, Grigni and Papadimitriou [STOC 1998, J. ACM 2002]. They proved that recognizing map graphs is in $\mathsf{NP}$ by proving the existence of a witness. Soon later, Thorup [FOCS 1998] claimed that recognizi…
A class of graphs is $χ$-bounded if there exists a function $f:\mathbb N\rightarrow \mathbb N$ such that for every graph $G$ in the class and an induced subgraph $H$ of $G$, if $H$ has no clique of size $q+1$, then the chromatic number of $H$ is less than or equal to $f(q)$. We denote by $W_n$ the wheel graph on $n+1$ vertices. We show that the class of graphs having no vertex-minor isomorphic to $W_n$ is $χ$-bounded. This generalizes several previous results; $χ$-boundedness for circle graphs, for graphs having no $W_5$ vertex-minors, and for graphs having no fan vertex-minors.
The adjacency matrices of graphs form a special subset of the set of all integer symmetric matrices. The description of which graphs have all their eigenvalues in the interval [-2,2] (i.e., those having spectral radius at most 2) has been known for several decades. In 2007 we extended this classification to arbitrary integer symmetric matrices. In this paper we turn our attention to symmetrizable matrices. We classify the connected nonsymmetric but symmetrizable matrices which have entries in $\Z$ that are maximal with respect to having all their eigenvalues in [-2,2]. This includes a spectral…
A clique of a graph is a maximal set of vertices of size at least 2 that induces a complete graph. A $k$-clique-colouring of a graph is a colouring of the vertices with at most $k$ colours such that no clique is monochromatic. Défossez proved that the 2-clique-colouring of perfect graphs is a $Σ_2^P$-complete problem [J. Graph Theory 62 (2009) 139--156]. We strengthen this result by showing that it is still $Σ_2^P$-complete for weakly chordal graphs. We then determine a hierarchy of nested subclasses of weakly chordal graphs whereby each graph class is in a distinct complexity class, namely $Σ…
Let $a$ and $b$ be positive integers such that $a\leq b$ and $a\equiv b\pmod 2$. We say that $G$ has all $(a, b)$-parity factors if $G$ has an $h$-factor for every function $h: V(G) \rightarrow \{a,a+2,\ldots,b-2,b\}$ with $b|V(G)|$ even and $h(v)\equiv b\pmod 2$ for all $v\in V(G)$. In this paper, we prove that every graph $G$ with $n\geq 3(b+1)(a+b)$ vertices has all $(a,b)$-parity factors if $δ(G)\geq (b^2-b)/a$, and for any two nonadjacent vertices $u,v \in V(G)$, $\max\{d_G(u),d_G(v)\}\geq \frac{bn}{a+b}$. Moreover, we show that this result is best possible in some sense.
In this article, we study quantum automorphism groups of distance-transitive graphs. We show that the odd graphs, the Hamming graphs $H(n,3)$, the Johnson graphs $J(n,2)$ and the Kneser graphs $K(n,2)$ do not have quantum symmetry. We also give a table with the quantum automorphism groups of all cubic distance-transitive graphs. Furthermore, with one graph missing, we can now decide whether or not a distance-regular graph of order $\leq 20$ has quantum symmetry. Moreover, we prove that the Hoffman-Singleton graph has no quantum symmetry. On a final note, we present an example of a pair of grap…
For a graph $G = (V, E)$ embedded in the projective plane, let $\mathcal{F}(G)$ denote the set of faces of $G$. Then, $G$ is called a $C_n$-face-magic projective graph if there exists a bijection $f: V(G) \to \{1, 2, \dots, |V(G)|\}$ such that for any $F \in \mathcal{F}(G)$ with $F \cong C_n$, the sum of all the vertex labels around $C_n$ is a constant $S$. We consider the $m \times n$ grid graph, denoted by $\mathcal{P}_{m,n}$, embedded in the projective plane in the natural way. Let $m \geqslant 3$ and $n \geqslant 3$ be odd integers. It is known that the $C_4$-face-magic value of a $C_4$-fa…
On cubic graphs having the maximal coalition number
本文研究了立方图中最大联盟数的未决问题，通过构造一个满足 $C(G)=9$ 的无限立方图族，证明了具有最大联盟数的立方图的存在性。
An indecomposable flow $f$ on a signed graph $Σ$ is a nontrivial integral flow that cannot be decomposed into $f=f_1+f_2$, where $f_1,f_2$ are nontrivial integral flows having the same sign (both $\geq 0$ or both $\leq 0$) at each edge of $Σ$. This paper is to classify indecomposable flows into characteristic vectors of circuits and Eulerian cycle-trees --- a class of signed graphs having a kind of tree structure in which all cycles can be viewed as vertices of a tree. Moreover, each indecomposable flow other than circuit characteristic vectors can be further decomposed into a sum of certain h…
The competition number k(G) of a graph G is the smallest number k such that G together with k isolated vertices added is the competition graph of an acyclic digraph. A chordless cycle of length at least 4 of a graph is called a hole of the graph. The number of holes of a graph is closely related to its competition number as the competition number of a graph which does not contain a hole is at most one and the competition number of a complete bipartite graph $K_{\lfloor \frac{n}{2} \rfloor, \lceil \frac{n}{2} \rceil}$ which has so many holes that no more holes can be added is the largest among …
The classical Whitney's 2-Isomorphism Theorem describes the families of graphs having the same cycle matroid. In this paper we describe the families of graphs having the same truncated cycle matroid and prove, in particular, that every 3-connected graph, except for K4, is uniquely defined by its truncated cycle matroid.
Perfect state transfer on bi-Cayley graphs over abelian groups
本文针对任意有限阿贝尔群上的双凯莱图，建立了完美状态转移的充分必要条件，并由此统一导出多类群上凯莱图的已知与新结果，还构造了二面体群上连通非正规凯莱图存在完美状态转移的反例。
A strong edge-coloring of a graph $G=(V,E)$ is a partition of its edge set $E$ into induced matchings. We study bipartite graphs with one part having maximum degree at most $3$ and the other part having maximum degree $Δ$. We show that every such graph has a strong edge-coloring using at most $3 Δ$ colors. Our result confirms a conjecture of Brualdi and Quinn Massey ~\cite{[BQ]} for this class of bipartite graphs.
We consider nonregular graphs having precisely three distinct eigenvalues. The focus is mainly on the case of graphs having two distinct valencies and our results include constructions of new examples, structure theorems, valency constraints, and a classification of certain special families of such graphs. We also present a new example of a graph with three valencies and three eigenvalues of which there are currently only finitely many known examples.
The problem of finding upper bounds for minimal vertex number of graphs with a given automorphism group is addressed in this article for the case of cyclic $2$-groups. We show that for any natural $n\ge 2$ there is an undirected graph having $2^n+6$ vertices and automorphism group cyclic of order $2^n$. This confirms an upper bound claimed by other authors for minimal number of vertices of undirected graphs having automorphism group $\mathbb{Z}/2^n\mathbb{Z}$.
We prove that the properties of having small discrepancy and having small second eigenvalue are equivalent in Cayley graphs, extending a result of Kohayakawa, Rödl, and Schacht, who treated the abelian case. The proof relies on Grothendieck's inequality. As a corollary, we also prove that a similar result holds in all vertex-transitive graphs.
The $(n-\ell)$-deck of an $n$-vertex graph is the multiset of subgraphs obtained from it by deleting $\ell$ vertices. A family of $n$-vertex graphs is $\ell$-recognizable if every graph having the same $(n-\ell)$-deck as a graph in the family is also in the family. We prove that the family of $n$-vertex graphs having no cycles is $\ell$-recognizable when $n\ge2\ell+1$ (except for $(n,\ell)=(5,2)$). It is known that this fails when $n=2\ell$.
Symmetry in complex unit gain graphs and their spectra
本文研究复单位增益图的结构、谱与符号对称性之间的关系，并构造无穷多个切换互异、谱对称但非符号对称的图，推广了符号图中的存在性结论。
For a graph with $n$ vertices and $m$ edges, having Laplacian spectrum $μ_1, μ_2, \cdots,μ_n$ and signless Laplacian spectrum $μ^+_1,μ^+_2, \cdots,μ^+_n$, the Laplacian energy and signless Laplacian energy of $G$ are respectively, defined as $LE(G)=\sum_{i=1}^{n}|μ_i-\frac{2m}{n}|$ and $LE^+(G)=\sum_{i=1}^{n}|μ^+_i-\frac{2m}{n}|$. Two graphs $G_1$ and $G_2$ of same order are said to be $L$-equienergetic if $LE(G_1)=LE(G_2)$ and $Q$-equienergetic if $LE^{+}(G_1)=LE^{+}(G_2)$. The problem of constructing graphs having same Laplacian energy has been considered by Stevanovic for threshold graphs a…
A graph \textit{G} is a tuple (\textit{V}, \textit{E}), where \textit{V} is the vertex set, \textit{E} is the edge set. A reduced graph is a graph of deleting non-Hamiltonian edges and smoothing out the redundant vertices of degree 2 on an edge except for leaving only one vertex of degree 2. A 2-common (\textit{v}, \textit{0}) combination is a cycle set in which every pair of joint cycles \textit{A} and \textit{B} satisfies $|V(A)\cap V(B)|=2$ and $|E(A)\cap E(B)|=0$. In this paper, we investigate the cycle structure of 2-common (\textit{v}, \textit{0}) combination in reduced graphs, and give …
An interval graph is the intersection graph of a finite set of intervals on a line and a circular-arc graph is the intersection graph of a finite set of arcs on a circle. While a forbidden induced subgraph characterization of interval graphs was found fifty years ago, finding an analogous characterization for circular-arc graphs is a long-standing open problem. In this work, we study the intersection graphs of finite sets of arcs on a circle no three of which cover the circle, known as normal Helly circular-arc graphs. Those circular-arc graphs which are minimal forbidden induced subgraphs for…
For a class $\mathcal C$ of graphs, we define $\mathcal C$-edge-brittleness of a graph $G$ as the minimum $\ell$ such that the vertex set of $G$ can be partitioned into sets inducing a subgraph in $\mathcal C$ and there are $\ell$ edges having ends in distinct parts. We characterize classes of graphs having bounded $\mathcal C$-edge-brittleness for a class $\mathcal C$ of forests or a class $\mathcal C$ of graphs with no $K_4\setminus e$ topological minors in terms of forbidden obstructions. We also define $\mathcal C$-vertex-brittleness of a graph $G$ as the minimum $\ell$ such that the edge …
Approximate proof labeling schemes were introduced by \\Censor-Hillel, Paz and Perry \cite{CPP}. Roughly speaking, a graph property~$\cP$ can be verified by an approximate proof labeling scheme in constant-time if the vertices of a graph having the property can be convinced, in a short period of time not depending on the size of the graph, that they are having the property $\cP$ or at least they are not far from being having the property $\cP$. The main result of this paper is that bounded-degree planar graphs (and also outer-planar graphs, bounded genus graphs, knotlessly embeddable graphs et…
Isomorphic pastings and the two possible structures for a pair of graphs having the same deck
本文研究图的牌组同构问题，定义同构粘贴概念，证明一对同态图同构的充要条件，并重构两个图参数及新有向图族，为重构猜想和DARC提供反证法新思路。
Fractional revival, known as a quantum transport phenomenon, is essential for entanglement generation in quantum spin networks. The concept of fractional revival is a generalization of perfect state transfer and periodicity on graphs. In this paper, we propose a sufficient and necessary condition for abelian Cayley graphs having fractional revival between any two distinct vertices. With this characterization, two general constructions of abelian Cayley graphs having fractional revival is presented. Meanwhile, we establish several new families of abelian Cayley graphs admitting fractional reviv…
Perfect state transfer is significant in quantum communication networks. There are very few graphs having this property. So, it is useful to find some new graphs having perfect state transfer. A good way to construct new graphs is by forming NEPS. It is known that the graph $P_{3}$ exhibits perfect state transfer and so we investigate some NEPS of the path $P_{3}$. A sufficient condition is found for a NEPS of $P_{3}$ to have perfect state transfer. Using these NEPS, some other graphs are constructed having perfect state transfer. We also prove that for every $n\in \Nl\setminus \left\lbrace 1\…
We describe computational results about undirected graphs having $10$ vertices and automorphism group isomorphic to $\mathbb{Z}/4\mathbb{Z}$.
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