Brouwer's Conjecture Research Articles

Brouwer's Conjecture for the cartesian product of graphs

Let G be a simple, undirected graph with n vertices and m edges and Sk(G) the sum of its k largest Laplacian eigenvalues. Brouwer conjectured that Sk(G)≤m+(k+12) for any 1≤k≤n. We show that if G and H are simple undirected graphs satisfying Brouwer's Conjecture, then the cartesian product G□H also satisfies Brouwer's Conjecture.

Laplacian spectrum and Brouwer's conjecture of a graph

Laplacian spectrum and Brouwer's conjecture of a graph

A Proof of Brouwer's Toughness Conjecture

The toughness $t(G)$ of a connected graph $G$ is defined as $t(G)=\min\{\frac{|S|}{c(G-S)}\}$, in which the minimum is taken over all proper subsets $S\subset V(G)$ such that $c(G-S)>1$, where $c(G-S)$ denotes the number of components of $G-S$. Let $\lambda$ denote the second largest absolute eigenvalue of the adjacency matrix of a graph. For any connected $d$-regular graph $G$, it has been shown by Alon that $t(G)>\frac{1}{3}(\frac{d^2}{d\lambda+\lambda^2}-1)$, through which, Alon was able to show that for every $t$ and $g$ there are $t$-tough graphs of girth strictly greater than $g$, and thus disproved in a strong sense a conjecture of Chv\'atal on pancyclicity. Brouwer independently discovered a better bound $t(G)>\frac{d}{\lambda}-2$ for any connected $d$-regular graph $G$, while he also conjectured that the lower bound can be improved to $t(G)\ge \frac{d}{\lambda} - 1$. We confirm this conjecture.

Constraints on Brouwer's Laplacian spectrum conjecture

Brouwer's Conjecture states that, for any graph G, the sum of the k largest (combinatorial) Laplacian eigenvalues of G is at most |E(G)|+(k+12), 1≤k≤n. We present several interrelated results establishing Brouwer's conjecture ▪ for a wide range of graphs G and parameters k. In particular, we show that (1) ▪ is true for low-arboricity graphs, and in particular for planar G when k≥11; (2) ▪ is true whenever the variance of the degree sequence is not very high, generalizing previous results for G regular or random; (3) ▪ is true if G belongs to a hereditarily spectrally-bounded class and k is sufficiently large as a function of k, in particular k≥32n for bipartite graphs; (4) ▪ holds unless G has edge-edit distance <k2n=O(n3/2) from a split graph; (5) no G violates the conjectured upper bound by more than O(n5/4), and bipartite G by no more than O(n); and (6) ▪ holds for all k outside an interval of length O(n3/4). Furthermore, we show that a natural generalization of Brouwer's conjecture surprisingly is quite false: asymptotically almost surely, a uniform random signed complete graph violates the conjectured bound by Ω(n).

Brouwer's conjecture holds asymptotically almost surely

We show that for a sequence of random graphs Brouwer's conjecture holds true with probability tending to one as the number of vertices tends to infinity. Surprisingly, it was found that a similar statement holds true for weighted graphs with possible negative weights as well. For graphs with a fixed number of vertices, the result implies that there are constants C>0 and n0 such that if n≥n0 then among all 2(n2) graphs with n vertices, at least (1−exp⁡(−Cn))2(n2) graphs satisfy Brouwer's conjecture.

On the sum of signless Laplacian spectra of graphs

For a simple graph $G(V,E)$ with $n$ vertices, $m$ edges, vertex set $V(G)=\{v_1, v_2, \dots, v_n\}$ and edge set $E(G)=\{e_1, e_2,\dots, e_m\}$, the adjacency matrix $A=(a_{ij})$ of $G$ is a $(0, 1)$-square matrix of order $n$ whose $(i,j)$-entry is equal to 1 if $v_i$ is adjacent to $v_j$ and equal to 0, otherwise. Let $D(G)={diag}(d_1, d_2, \dots, d_n)$ be the diagonal matrix associated to $G$, where $d_i=\deg(v_i),$ for all $i\in \{1,2,\dots,n\}$. The matrices $L(G)=D(G)-A(G)$ and $Q(G)=D(G)+A(G)$ are respectively called the Laplacian and the signless Laplacian matrices and their spectra (eigenvalues) are respectively called the Laplacian spectrum ($L$-spectrum) and the signless Laplacian spectrum ($Q$-spectrum) of the graph $G$. If $0=\mu_n\leq\mu_{n-1}\leq\cdots\leq\mu_1$ are the Laplacian eigenvalues of $G$, Brouwer conjectured that the sum of $k$ largest Laplacian eigenvalues $S_{k}(G)$ satisfies $S_{k}(G)=\sum\limits_{i=1}^{k}\mu_i\leq m+{k+1 \choose 2}$ and this conjecture is still open. If $q_1,q_2, \dots, q_n$ are the signless Laplacian eigenvalues of $G$, for $1\leq k\leq n$, let $S^{+}_{k}(G)=\sum_{i=1}^{k}q_i$ be the sum of $k$ largest signless Laplacian eigenvalues of $G$. Analogous to Brouwer's conjecture, Ashraf et al. conjectured that $S^{+}_{k}(G)\leq m+{k+1 \choose 2}$, for all $1\leq k\leq n$. This conjecture has been verified in affirmative for some classes of graphs. We obtain the upper bounds for $S^{+}_{k}(G)$ in terms of the clique number $\omega$, the vertex covering number $\tau$ and the diameter of the graph $G$. Finally, we show that the conjecture holds for large families of graphs.

Further developments on Brouwer's conjecture for the sum of Laplacian eigenvalues of graphs

Let G be a simple graph with order n and size m and having Laplacian eigenvalues μ1,μ2,…,μn−1,μn=0 and let Sk(G)=∑i=1kμi be the sum of k largest Laplacian eigenvalues of G. Brouwer conjectured that Sk(G)≤m+(k+12), for all k=1,2,…,n. We obtain upper bounds for Sk(G), in terms of the clique number ω, the order n and integers p≥0,r≥1,s1≥s2≥2 associated to the structure of the graph G. We discuss Brouwer's conjecture for two large families of graphs; the first family of graphs is obtained from a clique of size ω by identifying each of its vertices to a vertex of an arbitrary c-cyclic graph, and the second family is composed of the graphs in which the removal of the edges of the maximal complete bipartite subgraph gives a graph each of whose non-trivial components is a c-cyclic graph. We show among these two large families of graphs, the Brouwer's conjecture holds for various subfamilies of graphs depending upon the value of c, the order of the c-cyclic graphs, the clique number of the graph, the order of the maximal complete bipartite subgraph and the number of the c-cyclic components of the graph.

On Brouwer's conjecture for the sum of k largest Laplacian eigenvalues of graphs

Let G be a simple graph with n vertices. For 1≤k≤n, denote by Sk(G) the sum of k largest Laplacian eigenvalues of G. It is conjectured by Brouwer that Sk(G)≤e(G)+(k+12), where e(G) is the number of edges in G. This conjecture has been proved to be true for k=1, k=2, k=n−1, and k=n (for all graphs), and for certain classes of graphs such as trees, unicyclic graphs, and bicyclic graphs (for all k∈{1,2,…,n}). In this paper, we show that if Brouwer's conjecture is true for all graphs when k=p(1≤p≤(n−1)/2), then it is also true for all graphs when k=n−p−1, from which Brouwer's conjecture follows automatically for all graphs when k=n−2 and k=n−3. We also show that for a given graph G and its complement G‾, if Sk(G)≤e(G)+(k+12) holds for all k, then Sk(G‾)≤e(G‾)+(k+12) holds for all k as well, which implies that Brouwer's conjecture holds for the complements of trees, unicyclic graphs, and bicyclic graphs (for all k).

Improved results on Brouwer's conjecture for sum of the Laplacian eigenvalues of a graph

Let G be a graph with n vertices and m edges, and let Sk(G) be the sum of the k largest Laplacian eigenvalues of G. It was conjectured by Brouwer that Sk(G)≤m+(k+12) holds for 1≤k≤n. In this paper, we present several families of graphs for which Brouwer's conjecture holds, which improve some previously known results. We also establish a new upper bound on Sk(G) for split graphs, which is tight for each k∈{1,2,…,n−1} and turns out to be better than that conjectured by Brouwer.

Spectral threshold dominance, Brouwer's conjecture and maximality of Laplacian energy

The Laplacian energy of a graph is the sum of the distances of the eigenvalues of the Laplacian matrix of the graph to the graph's average degree. The maximum Laplacian energy over all graphs on n nodes and m edges is conjectured to be attained for threshold graphs. We prove the conjecture to hold for graphs with the property that for each k there is a threshold graph on the same number of nodes and edges whose sum of the k largest Laplacian eigenvalues exceeds that of the k largest Laplacian eigenvalues of the graph. We call such graphs spectrally threshold dominated. These graphs include split graphs and cographs and spectral threshold dominance is preserved by disjoint unions and taking complements. We conjecture that all graphs are spectrally threshold dominated. This conjecture turns out to be equivalent to Brouwer's conjecture concerning a bound on the sum of the k largest Laplacian eigenvalues.

On the sum of the Laplacian eigenvalues of a graph and Brouwer's conjecture

For a simple graph G with n-vertices, m edges and having Laplacian eigenvalues μ1,μ2,…,μn−1,μn=0, let Sk(G)=∑i=1kμi, be the sum of k largest Laplacian eigenvalues of G. Brouwer conjectured that Sk(G)≤m+(k+12), for all k=1,2,…,n. We obtain upper bounds for Sk(G) in terms of the clique number ω, the vertex covering number τ and the diameter d of a graph G. We show that Brouwer's conjecture holds for certain classes of graphs. The Laplacian energy LE(G) of a graph G is defined as LE(G)=∑i=1n|μi−d‾|, where d‾=2mn is the average degree of G. We obtain an upper bound for the Laplacian energy LE(G) of a graph G in terms of the number of vertices n, the number of edges m, the vertex covering number τ and the clique number ω of the graph.

On the vertex connectivity of a relation in an association scheme

We prove that for a sufficiently closed association scheme, the Brouwer conjecture on the coincidence of the vertex connectivity and the degree of any its connected basis relation is true. Bibliography: 7 titles.

On a sequence of (j, ε)-normal approximations to π/4 and the Brouwer conjecture

On a sequence of (j, ε)-normal approximations to π/4 and the Brouwer conjecture