Related papers: The maximum spectral radius of irregular bipartite…
Let G be a graph with n vertices and mu(G) be the largest eigenvalue of the adjacency matrix of G. We study how large mu(G) can be when G does not contain cycles and paths of specified order. In particular, we determine the maximum spectral…
We characterize the r-graph with maximal p-spectral radius among the k-partite r-graphs of order n, and the 3-graph with maximal p-spectral radius among the k-chromatic 3-graphs of order n.
Shiu, Chan and Chang [On the spectral radius of graphs with connectivity at most $k$, J. Math. Chem., 46 (2009), 340-346] studied the spectral radius of graphs of order $n$ with $\kappa(G) \leq k$ and showed that among those graphs, the…
The problems of maximizing the spectral radius and the number of spanning trees in a class of bipartite graphs with certain degree constraints are considered. In both the problems, the optimal graph is conjectured to be a Ferrers graph.…
Let $A(G)$ and $D(G)$ be the adjacency and degree matrices of a simple graph $G$ on $n$ vertices, respectively. The \emph{$A_\alpha$-spectral radius} of $G$ is the largest eigenvalue of $A_\alpha (G)=\alpha D(G)+(1-\alpha)A(G)$ for a real…
This paper gives a tight upper bound on the spectral radius of the signless Laplacian of graphs of given order and clique number. More precisely, let G be a graph of order n, let A be its adjacency matrix, and let D be the diagonal matrix…
Let $\mathcal{G}(m,k)$ be the set of graphs with size $m$ and odd girth (the length of shortest odd cycle) $k$. In this paper, we determine the graph maximizing the spectral radius among $\mathcal{G}(m,k)$ when $m$ is odd. As byproducts, we…
We give lower bounds on the size and total size of clique partitions of a graph in terms of its spectral radius and minimum degree, and derive a spectral upper bound for the maximum number of edge-disjoint $t$-cliques. The extremal graphs…
A signed graph $(G,\sigma)$ is a graph $G$ together with an assignment $\sigma$ of either a positive sign or a negative sign to each edge. A signed graph is unbalanced if it contains a cycle with odd number of negative edges. The spectral…
For a positive integer $k\ge 1$, a graph $G$ is $k$-stepwise irregular ($k$-SI graph) if the degrees of every pair of adjacent vertices differ by exactly $k$. Such graphs are necessarily bipartite. Using graph products it is demonstrated…
A bisection in a graph is a cut in which the number of vertices in the two parts differ by at most 1. In this paper, we give lower bounds for the maximum weight of bisections of edge-weighted graphs with bounded maximum degree. Our results…
The spectral analogue of the Tur\'{a}n type problem for hypergraphs is to determine the maximum spectral radius for the hypergraphs of order $n$ that do not contain a given hypergraph. For the hypergraphs among the set of the connected…
The classical spectral Tur\'{a}n problem is to determine the maximum spectral radius of an $\mathcal{F}$-free graph of order $n$. Zhai and Wang [Linear Algebra Appl, 437 (2012) 1641-1647] determined the maximum spectral radius of…
In this paper, the study of extreme value bounds for topological indices is crucial for understanding their influence on trees and bipartite graphs. For integers $\alpha, p$ satisfying $1 \leq p \leq \alpha \leq \Delta - 3$, the minimum…
We study gaps in the spectra of the adjacency matrices of large finite cubic graphs. It is known that the gap intervals $(2 \sqrt{2},3)$ and $[-3,-2)$ achieved in cubic Ramanujan graphs and line graphs are maximal. We give constraints on…
In this paper we study the maximum value of the largest eigenvalue for simple bipartite graphs, where the number of edges is given and the number of vertices on each side of the bipartition is given. We state a conjectured solution, which…
In 2018, Dankelmann, Gao, and Surmacs [J. Graph Theory, 88(1): 5--17, 2018] established sharp bounds on the oriented diameter of a bridgeless undirected graph and a bridgeless undirected bipartite graph in terms of vertex degree. In this…
In this paper, we obtain the sharp upper and lower bounds for the spectral radius of a nonnegative irreducible matrix. We also apply these bounds to various matrices associated with a graph or a digraph, obtain some new results or known…
Let $G$ be a graph with adjacency matrix $A(G)$ and let $D(G)$ be a diagonal matrix of the degrees of $G$. In 2017, Nikiforov defined the $A_{\alpha}$-matrix of $G$ as \begin{equation*} A_{\alpha}(G)=\alpha G)+(1-\alpha)A(G),…
A signed graph is a graph in which every edge carries a $+$ or a $-$ sign. In this paper, we determine the signed graphs with maximum spectral radius among all unbalanced signed graphs with fixed order that contain neither negative…