arXiv subjects

Sebastian M. Cioabă

Publications and source records attributed to Sebastian M. Cioabă.

41 records · Page 3Linked to original sources

arXiv2013

Edge-disjoint spanning trees and eigenvalues of regular graphs

Partially answering a question of Paul Seymour, we obtain a sufficient eigenvalue condition for the existence of $k$ edge-disjoint spanning trees in a regular graph, when $k\in \{2,3\}$. More precisely, we show that if the second largest eigenvalue of a $d$-regular graph $G$ is less than $d-\frac{2k-1}{d+1}$, then $G$ contains at least $k$ edge-disjoint spanning trees, when $k\in \{2,3\}$. We construct examples of graphs that show our bounds are essentially best possible. We conjecture that the above statement is true for any $k<d/2$.

math.CO↗

arXiv2013

The spectrum and toughness of regular graphs

In 1995, Brouwer proved that the toughness of a connected $k$-regular graph $G$ is at least $k/λ-2$, where $λ$ is the maximum absolute value of the non-trivial eigenvalues of $G$. Brouwer conjectured that one can improve this lower bound to $k/λ-1$ and that many graphs (especially graphs attaining equality in the Hoffman ratio bound for the independence number) have toughness equal to $k/λ$. In this paper, we improve Brouwer's spectral bound when the toughness is small and we determine the exact value of the toughness for many strongly regular graphs attaining equality in the Hoffman ratio bound such as Lattice graphs, Triangular graphs, complements of Triangular graphs and complements of point-graphs of generalized quadrangles. For all these graphs with the exception of the Petersen graph, we confirm Brouwer's intuition by showing that the toughness equals $k/(-λ_{min})$, where $λ_{min}$ is the smallest eigenvalue of the adjacency matrix of the graph.

math.CO↗

arXiv2013

Disconnecting strongly regular graphs

In this paper, we show that the minimum number of vertices whose removal disconnects a connected strongly regular graph into non-singleton components, equals the size of the neighborhood of an edge for many graphs. These include blocks graphs of Steiner $2$-designs, many Latin square graphs and strongly regular graphs whose intersection parameters are at most a quarter of their valency.

math.CO↗

arXiv2013

The graphs with all but two eigenvalues equal to $\pm 1$

We determine all graphs whose adjacency matrix has at most two eigenvalues (multiplicities included) different from $\pm 1$ and decide which of these graphs are determined by their spectrum. This includes the so-called friendship graphs, which consist of a number of edge-disjoint triangles meeting in one vertex. It turns out that the friendship graph is determined by its spectrum, except when the number of triangles equals sixteen.

math.CO↗

arXiv2007

The spectral radius and the maximum degree of irregular graphs

Let $G$ be an irregular graph on $n$ vertices with maximum degree $Δ$ and diameter $D$. We show that Δ-λ_1>\frac{1}{nD} where $λ_1$ is the largest eigenvalue of the adjacency matrix of $G$. We also study the effect of adding or removing few edges on the spectral radius of a regular graph.

math.CO↗