We prove three conjectures regarding the maximization of spectral invariants over certain families of graphs. Our most difficult result is that the join of P 2 and P n−2 is the unique graph of maximum spectral radius over all planar graphs. This was conjectured by Boots and Royle in 1991 and independently by Cao and Vince in 1993. Similarly, we prove a conjecture of Cvetković and Rowlinson from 1990 stating that the unique outerplanar graph of maximum spectral radius is the join of a vertex and P n−1 . Finally, we prove a conjecture of Aouchiche et al from 2008 stating that a pineapple graph is the unique connected graph maximizing the spectral radius minus the average degree. To prove our theorems, we use the leading eigenvector of a purported extremal graph to deduce structural properties about that graph.
Let G be a graph with no isolated vertices. A k-coupon coloring of G is an assignment of colors from [k] := {1, 2, . . . , k} to the vertices of G such that the neighborhood of every vertex of G contains vertices of all colors from [k]. The maximum k for which a k-coupon coloring exists is called the coupon coloring number of G, and is denoted χ c (G). In this paper, we prove that
A graph on $2k+1$ vertices consisting of $k$ triangles which intersect in exactly one common vertex is called a $k-$friendship graph and denoted by $F_k$. This paper determines the graphs of order $n$ that have the maximum (adjacency) spectral radius among all graphs containing no $F_k$, for $n$ sufficiently large.
The distance matrix of a graph G is the matrix containing the pairwise distances between vertices. The distance eigenvalues of G are the eigenvalues of its distance matrix and they form the distance spectrum of G. We determine the distance spectra of halved cubes, double odd graphs, and Doob graphs, completing the determination of distance spectra of distance regular graphs having exactly one positive distance eigenvalue. We characterize strongly regular graphs having more positive than negative distance eigenvalues. We give examples of graphs with few distinct distance eigenvalues but lacking regularity properties. We also determine the determinant and inertia of the distance matrices of lollipop and barbell graphs.
In this paper, we obtain two spectral upper bounds for the k-independence number of a graph which is is the maximum size of a set of vertices at pairwise distance greater than k. We construct graphs that attain equality for our first bound and show that our second bound compares favorably to previous bounds on the k-independence number.MSC: 05C50, 05C69
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.