The spectrum and toughness of regular graphs

In 1995, Brouwer proved that the toughness of a connected $k$-regular graph $G$ is at least $k/\lambda-2$, where $\lambda$ is the maximum absolute value of the non-trivial eigenvalues of $G$. Brouwer conjectured that one can improve this lower bound to $k/\lambda-1$ and that many graphs (especially graphs attaining equality in the Hoffman ratio bound for the independence number) have toughness equal to $k/\lambda$. 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 $k/(-\lambda_{min})$, where $\lambda_{min}$ is the smallest eigenvalue of the adjacency matrix of the graph.Comment: 15 pages, 1 figure, accepted to Discrete Applied Mathematics, special issue dedicated to the "Applications of Graph Spectra in Computer Science" Conference, Centre de Recerca Matematica (CRM), Bellaterra, Barcelona, June 16-20, 201

The least eigenvalue of the complements of graphs with given connectivity

The least eigenvalue of a graph $G$ is the least eigenvalue of adjacency matrix of $G$. In this paper we determine the graphs which attain the minimum least eigenvalue among all complements of connected simple graphs with given connectivity.Comment: 10 pages. arXiv admin note: substantial text overlap with arXiv:2209.0569

Relations between (κ, τ)-regular sets and star complements

Let G be a finite graph with an eigenvalue μ of multiplicity m. A set X of m vertices in G is called a star set for μ in G if μ is not an eigenvalue of the star complement G\X which is the subgraph of G induced by vertices not in X. A vertex subset of a graph is (k ,t)-regular if it induces a k -regular subgraph and every vertex not in the subset has t neighbors in it. We investigate the graphs having a (k,t)-regular set which induces a star complement for some eigenvalue. A survey of known results is provided and new properties for these graphs are deduced. Several particular graphs where these properties stand out are presented as examples

The zero forcing polynomial of a graph

Zero forcing is an iterative graph coloring process, where given a set of initially colored vertices, a colored vertex with a single uncolored neighbor causes that neighbor to become colored. A zero forcing set is a set of initially colored vertices which causes the entire graph to eventually become colored. In this paper, we study the counting problem associated with zero forcing. We introduce the zero forcing polynomial of a graph $G$ of order $n$ as the polynomial $\mathcal{Z}(G;x)=\sum_{i=1}^n z(G;i) x^i$, where $z(G;i)$ is the number of zero forcing sets of $G$ of size $i$. We characterize the extremal coefficients of $\mathcal{Z}(G;x)$, derive closed form expressions for the zero forcing polynomials of several families of graphs, and explore various structural properties of $\mathcal{Z}(G;x)$, including multiplicativity, unimodality, and uniqueness.Comment: 23 page