Related papers: Chromatic Signed-Symmetric Functions of Signed Gra…
The spectra of signed matrices have played a fundamental role in social sciences, graph theory, and control theory. In this work, we investigate the computational problems of identifying symmetric signings of matrices with natural spectral…
A signed graph $(G,\sigma)$ consists of a graph $G$ and the signature $\sigma : E(G) \rightarrow \{+1,-1\}$. An incidence of $G$ is a pair $(v,e)$, where $v$ is one of the end vertices of an edge $e \in E(G)$. A proper $q$-edge coloring…
The entropy of a graph is a functional depending both on the graph itself and on a probability distribution on its vertex set. This graph functional originated from the problem of source coding in information theory and was introduced by J.…
If we consider previously introduced extensions of Stanley's chromatic symmetric function $X_{G}(x_1, x_2, \ldots)$ for a graph $G$ to elements in the algebra $\textsf{QSym}$ of quasisymmetric functions and in the algebra $\textsf{NCSym}$…
The chromatic polynomials are studied by several authors and have important applications in different frameworks, specially, in graph theory and enumerative combinatorics. The aim of this work is to establish some properties of the…
The paper studies edge-coloring of signed multigraphs and extends classical Theorems of Shannon and K\"onig to signed multigraphs. We prove that the chromatic index of a signed multigraph $(G,\sigma_G)$ is at most $\lfloor \frac{3}{2}…
We propose a conjectural counting formula for the coefficients of the chromatic symmetric function of unit interval graphs using reinforcement learning. The formula counts specific disjoint cycle-tuples in the graphs, referred to as…
The chromatic polynomial $P(G,x)$ of a graph $G$ of order $n$ can be expressed as $\sum\limits_{i=1}^n(-1)^{n-i}a_{i}x^i$, where $a_i$ is interpreted as the number of broken-cycle free spanning subgraphs of $G$ with exactly $i$ components.…
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…
A \emph{mixed graph} is a graph with directed edges, called arcs, and undirected edges. A $k$-coloring of the vertices is proper if colors from ${1,2,...,k}$ are assigned to each vertex such that $u$ and $v$ have different colors if $uv$ is…
Determining whether two graphs are isomorphic is an important and difficult problem in graph theory. One way to make progress towards this problem is by finding and studying graph invariants that distinguish large classes of graphs. Stanley…
A graph $H$ is common if the number of monochromatic copies of $H$ in a 2-edge-colouring of the complete graph $K_n$ is asymptotically minimised by the random colouring. We prove that, given $k,r>0$, there exists a $k$-connected common…
Vertex coloring of a graph $G$ with $n$-colors can be equivalently thought to be a graph homomorphism (edge preserving vertex mapping) of $G$ to the complete graph $K_n$ of order $n$. So, in that sense, the chromatic number $\chi(G)$ of $G$…
In this paper, we generalize the concept of complete coloring and achromatic number to 2-edge-colored graphs and signed graphs. We give some useful relationships between different possible definitions of such achromatic numbers and prove…
Stanley's theory of $(P,\omega)$-partitions is a standard tool in combinatorics. It can be extended to allow for the presence of a restriction, that is a given maximal value for partitions at each vertex of the poset, as was shown by Assaf…
We prove that the chromatic symmetric function of any $n$-vertex tree containing a vertex of degree $d\geq \log _2n +1$ is not $e$-positive, that is, not a positive linear combination of elementary symmetric functions. Generalizing this, we…
We compute an explicit $e$-positive formula for the chromatic symmetric function of a lollipop graph, $L_{m,n}$. From here we deduce that there exist countably infinite distinct $e$-positive, and hence Schur-positive, bases of the algebra…
A signed graph is a graph together with an assignment of signs to the edges. A closed walk in a signed graph is said to be positive (negative) if it has an even (odd) number of negative edges, counting repetition. Recognizing the signs of…
In 2015, Matthias Beck and his team developed a computer program in SAGE which efficiently determines the number of signed proper $k$-colorings for a given signed graph. In this article, we determine the number of different signatures on…
We relate the combinatorial definitions of the type $A_n$ and type $C_n$ Stanley symmetric functions, via a combinatorially defined "double Stanley symmetric function," which gives the type $A$ case at $(\mathbf{x},\mathbf{0})$ and gives…