Graphs with small spectral gap
نویسنده
چکیده
It is conjectured that connected graphs with given number of vertices and minimum spectral gap (i.e., the difference between their two largest eigenvalues) are double kite graphs. The conjecture is confirmed for connected graphs with at most 10 vertices, and, using variable neighbourhood metaheuristic, there is evidence that it is true for graphs with at most 15 vertices. Several spectral properties of double kite graphs are obtained, including the equations for their first two eigenvalues. No counterexamples to the conjecture are obtained. Some numerical computations and comparisons that indicate its correctness are also given. Next, 3 lower and 3 upper bounds on spectral gap are derived, and some spectral and structural properties of the graphs that minimize the spectral gap are given. At the end, it is shown that in connected graphs any double kite graph has a unique spectrum.
منابع مشابه
SIGNLESS LAPLACIAN SPECTRAL MOMENTS OF GRAPHS AND ORDERING SOME GRAPHS WITH RESPECT TO THEM
Let $G = (V, E)$ be a simple graph. Denote by $D(G)$ the diagonal matrix $diag(d_1,cdots,d_n)$, where $d_i$ is the degree of vertex $i$ and $A(G)$ the adjacency matrix of $G$. The signless Laplacianmatrix of $G$ is $Q(G) = D(G) + A(G)$ and the $k-$th signless Laplacian spectral moment of graph $G$ is defined as $T_k(G)=sum_{i=1}^{n}q_i^{k}$, $kgeqslant 0$, where $q_1$,$q_2$, $cdots$, $q_n$ ...
متن کاملEla Graphs with Small Spectral Gap∗
It is conjectured that connected graphs with given number of vertices and minimum spectral gap (i.e., the difference between their two largest eigenvalues) are double kite graphs. The conjecture is confirmed for connected graphs with at most 10 vertices, and, using variable neighbourhood metaheuristic, there is evidence that it is true for graphs with at most 15 vertices. Several spectral prope...
متن کاملSpanning Trees of Small Degree
In this paper we show that pseudo-random graphs contain spanning trees of maximum degree 3. More specifically, (n, d, λ)-graphs with sufficiently large spectral gap contain such spanning trees.
متن کاملOn the Spectral Gap for Laplacians on Metric Graphs
This note will describe some of recent developments in the spectral theory of quantum graphs. More precisely, we are going to discuss the spectral gap the distance between two lowest eigenvalues for the Laplace operator on metric graphs in connection to geometric and topological properties of the underlying graphs. Our approach can be generalized further to include Schrödinger operators with no...
متن کاملTHE SPECTRAL DETERMINATION OF THE MULTICONE GRAPHS Kw ▽ C WITH RESPECT TO THEIR SIGNLESS LAPLACIAN SPECTRA
The main aim of this study is to characterize new classes of multicone graphs which are determined by their signless Laplacian spectra. A multicone graph is defined to be the join of a clique and a regular graph. Let C and K w denote the Clebsch graph and a complete graph on w vertices, respectively. In this paper, we show that the multicone graphs K w ▽C are determined by their signless ...
متن کامل