On the α-spectral radius of graphs

Starting with the study of the Collatz-Sinogowitz and the Albertson graph irregularity indices the relationships between the irregularity of graphs and their spectral radius are investigated. We also use the graph irregularity index defined as Ir(G) = -, where and denote the maximum and minimum degrees of G. Our observations lead to the answer for a question posed by Hong in 1993. The problem concerning graphs with the smallest spectral radius can be formulated as follows: If G is a connected irregular graph with n vertices and m edges, and G has the smallest spectral radius, is it true that Ir(G) =1? It will be shown that the answer is negative; counterexamples are represented by several cyclic graphs. Based on the previous considerations the problem proposed by Hong can be reinterpreted (refined) in the form of the following conjecture: If G is a connected irregular graph with n vertices and m edges, and G has the smallest spectral radius, then Ir(G)=1 if such a graph exists, and if not, then Ir(G)=2. Considering the family of unicyclic graphs for which Ir(G) 2, we prove that among n-vertex irregular unicyclic graphs the minimal spectral radius belongs to the uniquely defined short lollipop graphs where a pendent vertex is attached to cycle Cn-1. Moreover, it is verified that among n-vertex graphs there exists exactly one irregular graph Jn having a maximal spectral radius and an irregularity index of Ir(Jn)=1. Finally, it is also shown that by using the irregularity index Ir(G) a classification of n-vertex trees into (n-2) disjoint subsets can be performed.

Let GX be a graph obtained from a simple graph G by attaching a self-loop at each vertex of X ⊆ V ( G ) . The general extended adjacency matrix for the graph GX is defined and the bounds for the degree based energy of the graph GX are obtained. The study extends the notion of degree based energy of simple graphs to graphs with self-loops. For the graph GX of order n and size m with σ self-loops, the adjacency energy, E ( G X ) ≥ 4 mn + 2 σ ( n − σ ) n . The spectral radius ρ ( G X ) of its adjacency matrix is always less than or equal to 1 2 [ − 1 + 1 + 4 ( 2 m + σ + Δ − 1 ) ] , where Δ is the maximum degree in the graph GX and the equality conditions are given for 0 < σ < n . Few more bounds for ρ ( G X ) are also obtained. The study shows that, the spectral radius f max ( G X ) of its extended adjacency matrix satisfies: min i ∼ j F ij ( 2 m + σ ) n ≤ f max ( G X ) ≤ max i ∼ j F ij 2 ( 2 m + σ ) . We conclude the article by computing the extended adjacency spectrum of complete graph and complete bipartite graphs with self-loops.

Convex and quasiconvex functions on trees and their applications

A graph is said to be (?,?)-bidegreed if vertices all have one of two possible degrees: the maximum degree ? or the minimum degree ?, with ? ? ?. We show that in the set of connected (?,1)- bidegreed graphs, other than trees, with prescribed degree sequence, the graphs minimizing the adjacency matrix spectral radius cannot have vertices adjacent to ? - 1 vertices of degree 1, that is, there are not any hanging trees. Further we consider the limit point for the spectral radius of (?,1)-bidegreed graphs when degree ? vertices are inserted in each edge between any two degree ? vertices.

The spectral radius of edge chromatic critical graphs

The spectral radius of a graph is dened by that of its unoriented Laplacian matrix. In this paper, we determine the unicyclic graphs respectively with the third and the fourth largest spectral radius among all unicyclic graphs of given order.

Let $G$ be an irregular graph on $n$ vertices with maximum degree $\Delta$ and diameter $D$. We show that $$ \Delta-\lambda_1&gt;{1\over nD}, $$ where $\lambda_1$ 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.

Connected hypergraphs with small spectral radius

On the spectral radii and the signless Laplacian spectral radii of c-cyclic graphs with fixed maximum degree

C ‡ Abstract. Distance energy is a newly introduced molecular graph-based analog of the total �-electron energy, and it is defined as the sum of the absolute eigenvalues of the molecular distance matrix. For trees and unicyclic graphs, distance energy is equal to the doubled value of the distance spectral radius. In this paper, we introduce a general transformation that increases the distance spectral radius and provide an alternative proof that the path Pn has the maximal distance spectral radius among trees on n vertices. Among the trees with a fixed maximum degree �, we prove that the broom Bn,� (consisting of a star S�+1 and a path of length n � 1 attached to an arbitrary pendent vertex of the star) is the unique tree that maximizes the distance spectral radius, and conjecture the structure of a tree which minimizes the distance spectral radius. As a first step towards this conjecture, we characterize the starlike trees with the minimum distance spectral radius.

The spectral radius of a network graph is the largest Eigen value of the adjacency matrix of the graph. We hypothesize the spectral radius to be a measure of the variation in the degrees of the nodes. In this pursuit, we define a metric called the spectral radius ratio for node degree as the ratio of the spectral radius to the average node degree. We validate our hypothesis by determining this metric on some of the commonly studied classical large real-world complex network graphs (undirected) for network analysis. Based on the results collected, we observe the spectral radius ratio for node degree to be positively correlated (correlation coefficient: 0.75) to the coefficient of variation in node degree (the ratio of the average node degree to the standard deviation in node degree), thus confirming our hypothesis.

On the spectral radius of unicyclic graphs with fixed diameter

This paper investigates the relationship between spectral graph theory and graph properties, specifically focusing on the spectral radius, which is the largest eigenvalue of the adjacency matrix of a graph. Our problem is finding sharp upper bounds for the spectral radius of bipartite graphs with given bounde vertex sets. We first review existing inequalities, noting and discuss their limitations, noting particularly that some, like Hong's inequality, are not always sharp for all graph types. Our primary contribution is an elementary and direct approach to solving an optimization problem: finding a bipartite graph with a bounded number of vertices that maximizes its spectral radius. We prove that for any bipartite graph with bounded vertex sets of \(n_1\) and \(n_2\), the spectral radius \(\rho(G)\) is bounded by \(\sqrt{n_1n_2}\). We demonstrate that this inequality is sharp, with equality holding exclusively for the complete bipartite graph \(K_{n_1,n_2}\).

For a simple connected graph $G$, the convex linear combinations $D_{\alpha}(G)$ of \ $Tr(G)$ and $D(G)$ is defined as $D_{\alpha}(G)=\alpha Tr(G)+(1-\alpha)D(G)$, $0\leq \alpha\leq 1$. As $D_{0}(G)=D(G)$, $2D_{\frac{1}{2}}(G)=D^{Q}(G)$, $D_{1}(G)=Tr(G)$ and $D_{\alpha}(G)-D_{\beta}(G)=(\alpha-\beta)D^{L}(G)$, this matrix reduces to merging the distance spectral and distance signless Laplacian spectral theories. In this paper, we study the spectral properties of the generalized distance matrix $D_{\alpha}(G)$. We obtain some lower and upper bounds for the generalized distance spectral radius, involving different graph parameters and characterize the extremal graphs. Further, we obtain upper and lower bounds for the maximal and minimal entries of the $ p $-norm normalized Perron vector corresponding to spectral radius $ \partial(G) $ of the generalized distance matrix $D_{\alpha}(G)$ and characterize the extremal graphs.

For $ \alpha \in [0,1]$, let $A_{\alpha}(G) = \alpha D(G) +(1-\alpha)A(G)$ be $A_{\alpha}$-matrix, where $A(G)$ is the adjacent matrix and $D(G)$ is the diagonal matrix of the degrees of a graph $G$. Clearly, $A_{0} (G)$ is the adjacent matrix and $2 A_{\frac{1}{2}}$ is the signless Laplacian matrix. A connected graph is a cactus graph if any two cycles of $G$ have at most one common vertex. We first propose the result for subdivision graphs, and determine the cacti maximizing $A_{\alpha}$-spectral radius subject to fixed pendant vertices. In addition, the corresponding extremal graphs are provided. As consequences, we determine the graph with the $A_{\alpha}$-spectral radius among all the cacti with $n$ vertices; we also characterize the $n$-vertex cacti with a perfect matching having the largest $A_{\alpha}$-spectral radius.