相关论文: Vertex-Colored Graphs, Bicycle Spaces and Mahler M…

The (torsion) complexity of a finite edge-weighted graph is defined to be the order of the torsion subgroup of the abelian group presented by its Laplacian matrix. When G is d-periodic (i.e., G has a free action of the rank-d free abelian…

The (torsion) complexity of a finite signed graph is defined to be the order of the torsion subgroup of the abelian group presented by its Laplacian matrix. When $G$ is $d$-periodic (i.e., $G$ has a free ${\mathbb Z}^d$-action by graph…

For a graph $G$, a $k$-coloring $c:V(G)\to \{1,2,\ldots, k\}$ is called distinguishing, if the only automorphism $f$ of $G$ with the property $c(v)=c(f(v))$ for every vertex $v\in G$ (color-preserving automorphism), is the identity. In this…

The chromatic polynomial $\pi_{G}(k)$ of a graph $G$ can be viewed as counting the number of vertices in a family of coloring graphs $\mathcal C_k(G)$ associated with (proper) $k$-colorings of $G$ as a function of the number of colors $k$.…

For any given graph G = (V,E) we define in a certain way a new graph G(x,y,z) with the vertex set V\cup E depending on parameters x,y,z from {0,1, +, -} and call graph G(x,y,z) the (x,y,z)-transformation of G. It turns out that if G is an…

We introduce a new graph invariant \Lambda(G) that we call maxmaxflow, and put it in the context of some other well-known graph invariants, notably maximum degree and its relatives. We prove the equivalence of two "dual" definitions of…

The number of homomorphisms from a finite graph $F$ to the complete graph $K_n$ is the evaluation of the chromatic polynomial of $F$ at $n$. Suitably scaled, this is the Tutte polynomial evaluation $T(F;1-n,0)$ and an invariant of the cycle…

The distinguishing number $D(G)$ of a graph $G$ is the smallest number of colors that is needed to color the vertices of $G$ such that the only color preserving automorphism is the identity. For infinite graphs $D(G)$ is bounded by the…

A 2-edge-coloured graph $G$ is called {\bf locally complete} if for each vertex $v$, the vertices adjacent to $v$ through edges of the same colour induce a complete subgraph in $G$. Locally complete 2-edge-coloured graphs have nice…

A topological index of a graph $G$ is a real number which is preserved under isomorphism. Extensive studies on certain polynomials related to these topological indices have also been done recently. In a similar way, chromatic versions of…

We study the automorphisms of a Cayley graph that preserve its natural edge-colouring. More precisely, we are interested in groups G, such that every such automorphism of every connected Cayley graph on G has a very simple form: the…

Motivated by the definition of linear coloring on simplicial complexes, recently introduced in the context of algebraic topology \cite{Civan}, and the framework through which it was studied, we introduce the linear coloring on graphs. We…

In this paper, we consider the mixed tensor space of a $G$-graded vector space where $G$ is a finite abelian group. We obtain a spanning set of invariants of the associated symmetric algebra under the action of a color analogue of the…

If G is a graph and H is a set of subgraphs of G, then an edge-coloring of G is called H-polychromatic if every graph from H gets all colors present in G on its edges. The H-polychromatic number of G, denoted poly_H(G), is the largest…

In a graph G, cardinality of the smallest ordered set of vertices that distinguishes every element of V (G) is the (vertex) metric dimension of G. Similarly, the cardinality of such a set is the edge metric dimension of G, if it…

The author's recent results on spectral invariant dense subalgebras of C*-algebras associated with dynamical systems are summarized. If G is a compactly generated polynomial growth Type R Lie group, and the action of G on S(M) (Schwartz…

Let $P(G,\lambda)$ denote the number of proper vertex colorings of $G$ with $\lambda$ colors. The chromatic polynomial $P(C_n,\lambda)$ for the cycle graph $C_n$ is well-known as $$P(C_n,\lambda) = (\lambda-1)^n+(-1)^n(\lambda-1)$$ for all…

We present two new contributions to the study of the independence polynomial $Z_G(z)$ of a finite simple graph $G = (V,E)$. First, we provide an improved lower bound for the zero-free region of $Z_G(z)$ for the important class of claw-free…

Despite the fact that some vertex coloring problems are polynomially solvable on certain graph classes, most of these problems are not "under control" from a polyhedral point of view. The equivalence between \emph{optimization} and…

We consider vertex coloring of an acyclic digraph $\Gdag$ in such a way that two vertices which have a common ancestor in $\Gdag$ receive distinct colors. Such colorings arise in a natural way when bounding space for various genetic data…