Related papers: A note on skew spectrum of graphs
The positive and negative square energies of a graph, $s^+(G)$ and $s^-(G)$, are the sums of squares of the positive and negative eigenvalues of the adjacency matrix, respectively. The first results on square energies revealed symmetry…
Building on prior work that established Matrix Quasi-tree Theorems for special embedded graphs, in this paper, we develop a comprehensive theory applicable to all embedded graphs. We introduce symbolic skew-adjacency matrices and reduction…
The utility of a matrix satisfying the Strong Spectral Property has been well established particularly in connection with the inverse eigenvalue problem for graphs. More recently the class of graphs in which all associated symmetric…
The $t$-e.c. and pseudo-random property are typical properties of random graphs. In this note, we study the gap between them which has not been studied well. As a main result, we give the first explicit construction of infinite families of…
In this paper, we study the flip graph on the perfect matchings of a complete graph of even order. We investigate its combinatorial and spectral properties including connections to the signed reversal graph and we improve a previous upper…
A general novel approach mapping discrete, combinatorial, graph-theoretic problems onto ``physical'' models - namely $n$ simplexes in $n-1$ dimensions - is applied to the graph equivalence problem. It is shown to solve this long standing…
A graph $G$ is said to be \emph{determined by its spectrum} if any graph having the same spectrum as $G$ is isomorphic to $G$. Let $K_n \setminus P_{\ell}$ be the graph obtained from $K_n$ by removing edges of $P_\ell$, where $P_\ell$ is a…
Recent work has shown the utility of developing machine learning models that respect the structure and symmetries of eigenvectors. These works promote sign invariance, since for any eigenvector v the negation -v is also an eigenvector.…
A theory of orientation on gain graphs (voltage graphs) is developed to generalize the notion of orientation on graphs and signed graphs. Using this orientation scheme, the line graph of a gain graph is studied. For a particular family of…
It has been shown that the adjacency eigenspace of a network contains key information of its underlying structure. However, there has been no study on spectral analysis of the adjacency matrices of directed signed graphs. In this paper, we…
We consider non-trivial homomorphisms to reflexive oriented graphs in which some pair of adjacent vertices have the same image. Using a notion of convexity for oriented graphs, we study those oriented graphs that do not admit such…
This article deals with homomorphisms of oriented graphs with respect to push equivalence. Here homomorphisms refer to arc preserving vertex mappings, and push equivalence refers to the equivalence class of orientations of a graph $G$ those…
Clique-width is a well-studied graph parameter owing to its use in understanding algorithmic tractability: if the clique-width of a graph class ${\cal G}$ is bounded by a constant, a wide range of problems that are NP-complete in general…
We claimed that there is a polynomial algorithm to test if two graphs are isomorphic. But the algorithm is wrong. It only tests if the adjacency matrices of two graphs have the same eigenvalues. There is a counterexample of two…
Snake graphs appear naturally in the theory of cluster algebras. For cluster algebras from surfaces, each cluster variable is given by a formula which is parametrized by the perfect matchings of a snake graph. In this paper, we continue our…
We define smooth notions of concordance and sliceness for spatial graphs. We prove that sliceness of a spatial graph is equivalent to a condition on a set of linking numbers together with sliceness of a link associated to the graph. This…
This contribution extends linear models for feature vectors to sublinear models for graphs and analyzes their properties. The results are (i) a geometric interpretation of sublinear classifiers, (ii) a generic learning rule based on the…
This paper considers a hyperplane arrangement constructed with a subset of a set of all simple paths in a graph. A connection of the constructed arrangement to the maximum matching problem is established. Moreover, the problem of finding…
First, we derive explicit computable expressions of structured backward errors of approximate eigenelements of structured matrix polynomials including symmetric, skew-symmetric, Hermitian, skew-Hermitian, even and odd polynomials. We also…
We show that the set of $m \times m$ complex skew-symmetric matrix polynomials of odd grade $d$, i.e., of degree at most $d$, and (normal) rank at most $2r$ is the closure of the single set of matrix polynomials with the certain, explicitly…