Related papers: A horizontal-strip LLT polynomial is determined by…
It is well known that a graph is bipartite if and only if the spectrum of its adjacency matrix is symmetric. In the present paper, this assertion is dissected into three separate matrix results of wider scope, which are extended also to…
In this paper we show that Leavitt path algebras of weighted graphs and Leavitt path algebras of separated graphs are intimately related. We prove that any Leavitt path algebra $L(E,\omega)$ of a row-finite vertex weighted graph…
Color refinement is a classical technique used to show that two given graphs G and H are non-isomorphic; it is very efficient, although it does not succeed on all graphs. We call a graph G amenable to color refinement if it succeeds in…
We characterise when the Leavitt path algebras over $\mathbb{Z}$ of two arbitrary countable directed graphs are $*$-isomorphic by showing that two Leavitt path algebras over $\mathbb{Z}$ are $*$-isomorphic if and only if the corresponding…
Two $k$-uniform hypergraphs are said to be cospectral (E-cospectral), if their adjacency tensors have the same characteristic polynomial (E-characteristic polynomial). A $k$-uniform hypergraph $H$ is said to be determined by its spectrum,…
A graph $G$ is said to be \textit{determined by its generalized spectrum} (DGS for short) if for any graph $H$, $H$ and $G$ are cospectral with cospectral complements implies that $H$ is isomorphic to $G$. In \cite{WX,WX1}, Wang and Xu gave…
The chromatic functor of a simple graph is a functorization of the chromatic polynomial. M. Yoshinaga showed in \cite{Yoshinaga2015} that two finite graphs have isomorphic chromatic functors if and only if they have the same chromatic…
In the recent years, several polynomial algorithms of a dynamical nature have been proposed to address the graph isomorphism problem. In this paper we propose a generalization of an approach exposed in cond-mat/0209112 and find that this…
We show that the problem to decide whether two (convex) polytopes, given by their vertex-facet incidences, are combinatorially isomorphic is graph isomorphism complete, even for simple or simplicial polytopes. On the other hand, we give a…
Two isomorphic graphs can have inequivalent spatial embeddings in 3-space. In this way, an isomorphism class of graphs contains many spatial graph types. A common way to measure the complexity of a spatial graph type is to count the minimum…
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…
In a recent paper, we studied the interaction between the automorphism group of a graph and its Tutte polynomial. More precisely, we proved that certain symmetries of graphs are clearly reflected by their Tutte polynomials. The purpose of…
Weight-equitable partitions of graphs, which are a natural extension of the well-known equitable partitions, have been shown to be a powerful tool to weaken the regularity assumption in several well-known eigenvalue bounds. In this work we…
J. Przytycki has established a connection between the Hochschild homology of an algebra $A$ and the chromatic graph homology of a polygon graph with coefficients in $A$. In general the chromatic graph homology is not defined in the case…
The computational cost of simulating quantum many-body systems can often be reduced by taking advantage of physical symmetries. While methods exist for specific symmetry classes, a general algorithm to find the full permutation symmetry…
The Hamiltonian cycle polynomial can be evaluated to count the number of Hamiltonian cycles in a graph. It can also be viewed as a list of all spanning cycles of length $n$. We adopt the latter perspective and present a pair of original…
The graph isomorphism problem is considered. We assign modified $n$-variable characteristic polynomials for graphs and reduce the graph isomorphism problem to the problem of the polynomials isomorphism. It is required to find out, is there…
The solution of Shareshian-Wachs conjecture by Brosnan-Chow linked together the cohomology of regular semisimple Hessenberg varieties and graded chromatic symmetric functions on unit interval graphs. On the other hand, it is known that…
A graph $H$ is said to be positive if the homomorphism density $t_H(G)$ is non-negative for all weighted graphs $G$. The positive graph conjecture proposes a characterisation of such graphs, saying that a graph is positive if and only if it…
The LLT polynomials $\mathcal{L}_{\mathbf{\beta}/\mathbf{\gamma}} (X;t)$ are a family of symmetric polynomials indexed by a tuple of (possibly skew-)partitions $\mathbf{\beta}/\mathbf{\gamma}=…