Related papers: Graphs and matrices of maximal energy
We prove conditions for equality between the extreme eigenvalues of a matrix and its quotient. In particular, we give a lower bound on the largest singular value of a matrix and generalize a result of Finck and Grohmann about the largest…
If $\Gamma$ is a graph for which every edge is in exactly one clique of order $\omega$, then one can form a new graph with vertex set equal to these cliques. This is a generalization of the line graph of $\Gamma$. We discover many general…
Any gentle algebra $A$ with one maximal path corresponds to a unique quasi-diagram $\alpha$. We introduce the regularity for $\alpha$, and show that $A$ has finite global dimension if and only if $\alpha$ is regular. We characterize regular…
In this paper, we investigate certain graphs defined on groups, with a focus on infinite groups. The graphs discussed are the power graph, the enhanced power graph, and the commuting graph whose vertex set is a group $G$. The power graph is…
In this paper we address the graph matching problem. Following the recent works of \cite{zaslavskiy2009path,Vestner2017} we analyze and generalize the idea of concave relaxations. We introduce the concepts of conditionally concave and…
We study the energy function of the Kuramoto model in random geometric graphs defined in the unit circle as the number of nodes diverges. We prove the existence of at least one local minimum for each winding number $q \in \mathbb{Z}$ with…
We give necessary and sufficient conditions on the parameters of a regular graph $\Gamma$ (with or without loops) such that $E(\Gamma)=E(\overline \Gamma)$. We study complementary equienergetic cubic graphs obtaining classifications up to…
Characterized are all simple undirected graphs $G$ such that any real symmetric matrix that has graph $G$ has no eigenvalues of multiplicity more than 2. All such graphs are partial 2-trees (and this follows from a result for rather general…
A graph property is said to be elusive ( evasive) if every algorithm testing this property by asking questions of the form "is there an edge between vertices x and y" requires, in the worst case, to ask about all pairs of vertices. The…
We determine the maximum number of edges of an $n$-vertex graph $G$ with the property that none of its $r$-cliques intersects a fixed set $M\subset V(G)$. For $(r-1)|M|\ge n$, the $(r-1)$-partite Turan graph turns out to be the unique…
Numerical simulations of coupled map lattices (CMLs) and other complex model systems show an enormous phenomenological variety that is difficult to classify and understand. It is therefore desirable to establish analytical tools for…
We consider infinite graphs and the associated energy forms. We show that a graph is canonically compactifiable (i.e. all functions of finite energy are bounded) if and only if the underlying set is totally bounded with respect to any…
This work develops a flexible and mathematically sound framework for the design and analysis of graph scattering networks with variable branching ratios and generic functional calculus filters. Spectrally-agnostic stability guarantees for…
The energy of a graph was introduced by {\sc Gutman} in 1978 as the sum of the absolute values of the eigenvalues of its adjacency matrix. We study the energy of integral circulant graphs, also called gcd graphs, which can be characterized…
We consider the spherical limit of multi-matrix models on regular target graphs, for instance single or multiple Potts models, or lattices of arbitrary dimension. We show, to all orders in the low temperature expansion, that when the degree…
We prove that, for any graph $G$, its graph energy is at least twice the Randic index. We show that equality holds if and only if $G$ is the union of complete bipartite graphs.
Quantum graphs and their experimental counterparts, microwave networks, are ideally suited to study the spectral statistics of chaotic systems. The graph spectrum is obtained from the zeros of a secular determinant derived from energy and…
Given an assignment of weights w to the edges of a graph G, a matching M in G is called strongly w-maximal if for any matching N the sum of weights of the edges in N\M is at most the sum of weights of the edges in M\N. We prove that if w…
Let $G$ be a simple graph of order $n$ with degree sequence $(d)=(d_1,d_2,\ldots,d_n)$ and conjugate degree sequence $(d^*)=(d_1^*,d_2^*,\ldots,d_n^*)$. In \cite{AkbariGhorbaniKoolenObudi2010,DasMojallalGutman2017} it was proven that…
In this work we use matrix models to study the problem of strength distributions. This is motivated by noticing near exponential fall offs of strengths in calculated magnetic dipole excitations. We emphasize that the quality of the…