We give necessary and sufficient conditions under which the Jacobian of a graph is generated by a divisor that is the difference of two vertices. This answers a question posed by Becker and Glass and allows us to prove various other propositions about the order of divisors that are the difference of two vertices. We conclude with some conjectures about these divisors on random graphs and support them with empirical evidence.
For any integer $d \geq 1$, we verify the Jacobian Conjecture for a $d$-linear map in two variables. We prove that almost all the coefficients of the formal inverse are in the ideal specified by the Jacobian condition. We find expressions for certain elements in terms of the generators of this ideal. To obtain these expressions, we generalize a bijective proof of the Cayley-Hamilton theorem in two ways.
We consider a family of I-graphs I(n,k,l), which is a generalization of the class of generalized Petersen graphs. In the present paper, we provide a new method for counting Jacobian group of the I-graph I(n,k,l). We show that the minimum number of generators of Jac(I(n,k,l)) is at least two and at most 2k + 2l - 1. Also, we obtain a closed formula for the number of spanning trees of I(n,k,l) in terms of Chebyshev polynomials. We investigate some arithmetical properties of this number and its asymptotic behaviour.
This paper proves results about the Jacobians of a certain family of covering graphs, $Y$, of a base graph $X$, that is constructed from an assignment of elements from a group $G$ to the edges of $X$ ($G$ is called the voltage group and $Y$ is called the derived graph). Of particular interest is when the voltage assignment is given by mapping a generator of the cyclic group of order $d$ to a single edge of $X$ (all other edges are assigned the identity), called a single voltage assignment. Both the order and abelian group structure of the Jacobian of single voltage assignment derived graphs ar…
Let $G$ be a connected graph. A vertex $w\in V(G)$ strongly resolves two vertices $u,v\in V(G)$ if there exists some shortest $u-w$ path containing $v$ or some shortest $v-w$ path containing $u$. A set $S$ of vertices is a strong metric generator for $G$ if every pair of vertices of $G$ is strongly resolved by some vertex of $S$. The smallest cardinality of a strong metric generator for $G$ is called the strong metric dimension of $G$. It is well known that the problem of computing this invariant is NP-hard. In this paper we study the problem of finding exact values or sharp bounds for the str…
We introduce a monomial ideal whose standard monomials encode the vertices of all fibers of a lattice. We study the minimal generators, the radical, the associated primes and the primary decomposition of this ideal, as well as its relation to initial ideals of lattice ideals.
In a graph G; a vertex (resp. an edge) metric generator is a set of vertices S such that any pair of vertices (resp. edges) from G is distinguished by at least one vertex from S: The cardinality of a smallest vertex (resp. edge) metric generator is the vertex (resp. edge) metric dimension of G: In [19] we determined the vertex (resp. edge) metric dimension of unicyclic graphs and that it takes its value from two consecutive integers. Therein, several cycle configurations were introduced and the vertex (resp. edge) metric dimension takes the greater of the two consecutive values only if any of …
Let G be a simple graph and let J be its ideal of vertex covers. We give a graph theoretical description of the irreducible b-vertex covers of G, i.e., we describe the minimal generators of the symbolic Rees algebra of J. Then we study the irreducible b-vertex covers of the blocker of G, i.e., we study the minimal generators of the symbolic Rees algebra of the edge ideal of G. We give a graph theoretical description of the irreducible binary b-vertex covers of the blocker of G. It is shown that they correspond to irreducible induced subgraphs of G. As a byproduct we obtain a method, using Hilb…
J. Herzog, T. Hibi, N. V. Trung and X. Zheng characterize the vertex cover algebras which are standard graded. In this paper we give a simple combinatorial criterion for the standard graded property of vertex cover algebras in the case of quasi-trees. We also give an example of how this criterion works and compute the maximal degree of a minimal generator in that case.
A vertex $v\in V(G)$ is said to distinguish two vertices $x,y\in V(G)$ of a nontrivial connected graph $G$ if the distance from $v$ to $x$ is different from the distance from $v$ to $y$. A set $S\subset V(G)$ is a local metric generator for $G$ if every two adjacent vertices of $G$ are distinguished by some vertex in $S$. A local metric generator with the minimum cardinality is called a local metric basis for $G$ and its cardinality, the local metric dimension of G. In this paper we study the problem of finding exact values for the local metric dimension of corona product of graphs.
Given a connected graph $G$, a vertex $w\in V(G)$ strongly resolves two vertices $u,v\in V(G)$ if there exists some shortest $u-w$ path containing $v$ or some shortest $v-w$ path containing $u$. A set $S$ of vertices is a strong metric generator for $G$ if every pair of vertices of $G$ is strongly resolved by some vertex of $S$. The smallest cardinality of a strong metric generator for $G$ is called the strong metric dimension of $G$. In this paper we obtain several relationships between the strong metric dimension of the lexicographic product of graphs and the strong metric dimension of its f…
Let $Γ=(V,E)$ be a simple connected graph. $d(α,ε)=min\{d(α, w), d(α, d\}$ computes the distance between a vertex $α\in V(Γ)$ and an edge $ε=wd\in E(Γ)$. A single vertex $α$ is said to recognize (resolve) two different edges $ε_{1}$ and $ε_{2}$ from $E(Γ)$ if $d(α, ε_{2})\neq d(α, ε_{1}\}$. A subset of distinct ordered vertices $U_{E}\subseteq V(Γ)$ is said to be an edge metric generator for $Γ$ if every pair of distinct edges from $Γ$ are recognized by some element of $U_{E}$. An edge metric generator with a minimum number of elements in it, is called an edge metric basis for $Γ$. Then, the c…
Let $G=(V, E)$ be a connected graph. Given a vertex $v\in V$ and an edge $e=uw\in E$, the distance between $v$ and $e$ is defined as $d_G(e,v)=\min\{d_G(u,v),d_G(w,v)\}$. A nonempty set $S\subset V$ is an edge metric generator for $G$ if for any two edges $e_1,e_2\in E$ there is a vertex $w\in S$ such that $d_G(w,e_1)\ne d_G(w,e_2)$. The minimum cardinality of any edge metric generator for a graph $G$ is the edge metric dimension of $G$. The edge metric dimension of the join, lexicographic and corona product of graphs is studied in this article.
Given a connected graph $G$, a vertex $w\in V(G)$ distinguishes two different vertices $u,v$ of $G$ if the distances between $w$ and $u$ and between $w$ and $v$ are different. Moreover, $w$ strongly resolves the pair $u,v$ if there exists some shortest $u-w$ path containing $v$ or some shortest $v-w$ path containing $u$. A set $W$ of vertices is a (strong) metric generator for $G$ if every pair of vertices of $G$ is (strongly resolved) distinguished by some vertex of $W$. The smallest cardinality of a (strong) metric generator for $G$ is called the (strong) metric dimension of $G$. In this art…
Let $G=(V,E)$ be a connected graph. A vertex $w\in V$ distinguishes two elements (vertices or edges) $x,y\in E\cup V$ if $d_G(w,x)\ne d_G(w,y)$. A set $S$ of vertices in a connected graph $G$ is a mixed metric generator for $G$ if every two elements (vertices or edges) of $G$ are distinguished by some vertex of $S$. The smallest cardinality of a mixed metric generator for $G$ is called the mixed metric dimension and is denoted by $\mathrm{mdim}(G)$. In this paper we consider the structure of mixed metric generators and characterize graphs for which the mixed metric dimension equals the trivial…
We derive a recurrence relation for the number of simple vertex-labelled bipartite graphs with given degrees of the vertices and use this result to obtain a new method for computing the growth function of the Artin monoid of type $A_{n-1}$ with respect to the simple elements (permutation braids) as generators. Instead of matrices of size $2^{n-1}\times 2^{n-1}$, we use matrices of size $p(n)\times p(n)$, where $p(n)$ is the number of partitions of $n$.
A vertex $v\in V(G)$ is said to distinguish two vertices $x,y\in V(G)$ of a nontrivial connected graph $G$ if the distance from $v$ to $x$ is different from the distance from $v$ to $y$. A set $S\subset V(G)$ is a local metric generator for $G$ if every two adjacent vertices of $G$ are distinguished by some vertex of $S$. A local metric generator with the minimum cardinality is called a local metric basis for $G$ and its cardinality, the local metric dimension of $G$. It is known that the problem of computing the local metric dimension of a graph is NP-Complete. In this paper we study the prob…
Let $G=(V,E)$ be a connected graph, let $v\in V$ be a vertex and let $e=uw\in E$ be an edge. The distance between the vertex $v$ and the edge $e$ is given by $d_G(e,v)=\min\{d_G(u,v),d_G(w,v)\}$. A vertex $w\in V$ distinguishes two edges $e_1,e_2\in E$ if $d_G(w,e_1)\ne d_G(w,e_2)$. A set $S$ of vertices in a connected graph $G$ is an edge metric generator for $G$ if every two edges of $G$ are distinguished by some vertex of $S$. The smallest cardinality of an edge metric generator for $G$ is called the edge metric dimension and is denoted by $edim(G)$. In this article we introduce the concept…
As a generalization of the concept of the partition dimension of a graph, this article introduces the notion of the $k$-partition dimension. Given a nontrivial connected graph $G=(V,E)$, a partition $Π$ of $V$ is said to be a $k$-partition generator for $G$ if any pair of different vertices $u,v\in V$ is distinguished by at least $k$ vertex sets of $Π$, \emph{i.e}., there exist at least $k$ vertex sets $S_1,\ldots,S_k\inΠ$ such that $d(u,S_i)\ne d(v,S_i)$ for every $i\in\{1,\ldots,k\}$. A $k$-partition generator for $G$ with minimum cardinality among all their $k$-partition generators is calle…
A vertex $w$ of a connected graph $G$ strongly resolves two vertices $u,v\in V(G)$, if there exists some shortest $u-w$ path containing $v$ or some shortest $v-w$ path containing $u$. A set $S$ of vertices is a strong metric generator for $G$ if every pair of vertices of $G$ is strongly resolved by some vertex of $S$. The smallest cardinality of a strong metric generator for $G$ is called the strong metric dimension of $G$. In this paper we obtain several tight bounds or closed formulae for the strong metric dimension of the Cartesian sum of graphs in terms of the strong metric dimension, cliq…
This paper gives lower bounds on the spectral radius of vertex-transitive graphs, based on the number of ``prime cycles'' at a vertex. The bounds are obtained by constructing circuits in the graph that resemble ``cactus trees'', and enumerating them. Counting these circuits gives a coefficient-wise underestimation of the Green function of the graph, and hence and underestimation of its spectral radius. The bounds obtained are very good for the Cayley graph of surface groups of genus g>=2, with standard generators (these graphs are the 1-skeletons of tessellations of hyperbolic plane by 4g-gons…
We show that the minimum number of vertices of a simplicial complex with fundamental group $\mathbb{Z}^{n}$ is at most $O(n)$ and at least $Ω(n^{3/4})$. For the upper bound, we use a result on orthogonal 1-factorizations of $K_{2n}$. For the lower bound, we use a fractional Sylvester-Gallai result. We also prove that any group presentation $\langle S | R\rangle \cong \mathbb{Z}^{n}$ whose relations are of the form $g^{a}h^{b}i^{c}$ for $g, h, i \in S$ has at least $Ω(n^{3/2})$ generators.
A vertex $v\in V$ is said to resolve two vertices $x$ and $y$ if $d_G(v,x)\ne d_G(v,y)$. A set $S\subset V$ is said to be a metric generator for $G$ if any pair of vertices of $G$ is resolved by some element of $S$. A minimum metric generator is called a metric basis, and its cardinality, $\dim(G)$, the \emph{metric dimension} of $G$. A set $S\subseteq V$ is said to be a simultaneous metric generator for a graph family ${\cal G}=\{G_1,G_2,\ldots,G_k\}$, defined on a common (labeled) vertex set, if it is a metric generator for every graph of the family. A minimum cardinality simultaneous metric…
Let ${\mathcal G}$ be a graph family defined on a common (labeled) vertex set $V$. A set $S\subseteq V$ is said to be a simultaneous metric generator for ${\cal G}$ if for every $G\in {\cal G}$ and every pair of different vertices $u,v\in V$ there exists $s\in S$ such that $d_{G}(s,u)\ne d_{G}(s,v)$, where $d_{G}$ denotes the geodesic distance. A simultaneous adjacency generator for ${\cal G}$ is a simultaneous metric generator under the metric $d_{G,2}(x,y)=\min\{d_{G}(x,y),2\}$. A minimum cardinality simultaneous metric (adjacency) generator for ${\cal G}$ is a simultaneous metric (adjacency…
We take a graph theoretic approach to the problem of finding generators for those prime ideals of $\mathcal{O}_q(\mathcal{M}_{m,n}(\mathbb{K}))$ which are invariant under the torus action ($\mathbb{K}^*)^{m+n}$. Launois \cite{launois3} has shown that the generators consist of certain quantum minors of the matrix of canonical generators of $\mathcal{O}_q(\mathcal{M}_{m,n}(\mathbb{K}))$ and in \cite{launois2} gives an algorithm to find them. In this paper we modify a classic result of Lindström \cite{lind} and Gessel-Viennot~\cite{gv} to show that a quantum minor is in the generating set for a p…
