Related papers: The energy of a simplicial complex

Let V be a finite dimensional complex superspace and G a simple (or a ``close'' to simple) Lie superalgebra of matrix type, i.e., a Lie subsuperalgebra in GL(V). Under the classical invariant theory for G we mean the description of…

The power graph $\mathscr{P}(G)$ of a group $G$ is defined as the simple graph with vertex set $G$, and where two distinct vertices $x$ and $y$ are joined by an edge if and only if either $x= y^k$ or $y= x^k$, $k \in \mathbb{N}$. Here we…

A classical theorem of R. Baer describes the nilpotent radical of a finite group G as the set of all Engel elements, i.e. elements y in G such that for any x in G the n-th commutator [x,y,...,y] equals 1 for n big enough. We obtain a…

We describe a positive energy theorem for Einstein gravity coupled to scalar fields with first-derivative interactions, so-called P(X,phi) theories. We offer two independent derivations of this result. The first method introduces an…

The energy of a simple graph $G$, denoted by $E(G)$, is defined as the sum of the absolute values of all eigenvalues of its adjacency matrix. Denote by $C_n$ the cycle, and $P_n^{6}$ the unicyclic graph obtained by connecting a vertex of…

Let $\Gamma$ be a simple graph with $n$ vertices. The energy of $\Gamma$, denoted by $\mathcal{E}(\Gamma)$, is defined as the sum of the absolute values of the eigenvalues of $\Gamma$. The graph $\Gamma$ is said to be hyperenergetic if…

Let $G$ be a simple undirected $n$-vertex graph with the characteristic polynomial of its Laplacian matrix $L(G)$, $\det (\lambda I - L (G))=\sum_{k = 0}^n (-1)^k c_k \lambda^{n - k}$. Laplacian--like energy of a graph is newly proposed…

Given a simplicial complex $X$, we construct a simplicial complex $\Omega X$ that may be regarded as a combinatorial version of the based loop space of a topological space. Our construction explicitly describes the simplices of $\Omega X$…

Let $G$ be a graph of order $n$, and $d_i$ the degree of a vertex $v_i$ of $G$. The Randi\'c matrix ${\bf R}=(r_{ij})$ of $G$ is defined by $r_{ij} = 1 / \sqrt{d_jd_j}$ if the vertices $v_i$ and $v_j$ are adjacent in $G$ and $r_{ij}=0$…

Let G be a connected reductive group. We define a map from the set of unipotent classes in G to the set of conjugacy classes in the Weyl group (assuming that the characteristic is not bad). This map is a one sided inverse of a map in the…

There are several graphs defined on groups. Among them we consider graphs whose vertex set consists conjugacy classes of a group $G$ and adjacency is defined by properties of the elements of conjugacy classes. In particular, we consider…

The incidence energy $\mathscr{IE}(G)$ of a graph $G$, defined as the sum of the singular values of the incidence matrix of a graph $G$, is a much studied quantity with well known applications in chemical physics. The Laplacian-energy-like…

Let $G$ be a simple connected graph on $n$ vertices, and let $\lambda_1(G),\lambda_2(G),\ldots,\lambda_n(G)$ be the eigenvalues of its adjacency matrix $A(G)$. For $p>0$, define the $p$-energy of $G$ by $\mathcal E_p(G)=\sum_{i=1}^n…

The energy of a graph is defined as the sum of the absolute values of the eigenvalues of its adjacency matrix. In this paper, we characterize the tetracyclic graph of order $n$ with minimal energy. By this, the validity of a conjecture for…

We introduce generalized energies for a class of U_q(D^{(1)}_n) crystals by using the piecewise linear functions that are building blocks of the combinatorial R. They include the conventional energy in the theory of affine crystals as a…

Let $G$ be a finite group, and let $x$ be an element of $G$. Denote by $\Sol_G(x)$ the set of all $y \in G$ such that the group generated by $x$ and $y$ is soluble. We investigate the influence of $\Sol_G(x)$ on the structure of $G$.

We establish a weak-strong uniqueness result for the isentropic compressible Euler equations, that is: As long as a sufficiently regular solution exists, all energy-admissible weak solutions with the same initial data coincide with it. The…

The free energy of a multi-component scalar field theory is considered as a functional W[G,J] of the free correlation function G and an external current J. It obeys non-linear functional differential equations which are turned into…

The general problem of representing the Greens function $G(k,z)$ in terms of self energy, in field theories lacking Wick's theorem, is considered. A simple construction shows that a Dyson like representation with a self energy $\Sigma(k,z)$…

Let M and N be even-dimensional oriented real manifolds, and $u:M \to N$ be a smooth mapping. A pair of complex structures at M and N is called u-compatible if the mapping u is holomorphic with respect to these structures. The quotient of…