On the maximum AαA_{\alpha}-spectral radius of unicyclic and bicyclic graphs with fixed girth or fixed number of pendant vertices

For a connected graph $G$, let $A(G)$ be the adjacency matrix of $G$ and $D(G)$ be the diagonal matrix of the degrees of the vertices in $G$. The $A_{\alpha}$-matrix of $G$ is defined as \begin{align*} A_\alpha (G) = \alpha D(G) + (1-\alpha) A(G) \quad \text{for any $\alpha \in [0,1]$}. \end{align*} The largest eigenvalue of $A_{\alpha}(G)$ is called the $A_{\alpha}$-spectral radius of $G$. In this article, we characterize the graphs with maximum $A_{\alpha}$-spectral radius among the class of unicyclic and bicyclic graphs of order $n$ with fixed girth $g$. Also, we identify the unique graphs with maximum $A_{\alpha}$-spectral radius among the class of unicyclic and bicyclic graphs of order $n$ with $k$ pendant vertices.Comment: 16 page

The inertia of weighted unicyclic graphs

Let $G_w$ be a weighted graph. The \textit{inertia} of $G_w$ is the triple $In(G_w)=\big(i_+(G_w),i_-(G_w),$ $i_0(G_w)\big)$, where $i_+(G_w),i_-(G_w),i_0(G_w)$ are the number of the positive, negative and zero eigenvalues of the adjacency matrix $A(G_w)$ of $G_w$ including their multiplicities, respectively. $i_+(G_w)$, $i_-(G_w)$ is called the \textit{positive, negative index of inertia} of $G_w$, respectively. In this paper we present a lower bound for the positive, negative index of weighted unicyclic graphs of order $n$ with fixed girth and characterize all weighted unicyclic graphs attaining this lower bound. Moreover, we characterize the weighted unicyclic graphs of order $n$ with two positive, two negative and at least $n-6$ zero eigenvalues, respectively.Comment: 23 pages, 8figure

Eccentric connectivity index

The eccentric connectivity index $\xi^c$ is a novel distance--based molecular structure descriptor that was recently used for mathematical modeling of biological activities of diverse nature. It is defined as $\xi^c (G) = \sum_{v \in V (G)} deg (v) \cdot \epsilon (v)$\,, where $deg (v)$ and $\epsilon (v)$ denote the vertex degree and eccentricity of $v$\,, respectively. We survey some mathematical properties of this index and furthermore support the use of eccentric connectivity index as topological structure descriptor. We present the extremal trees and unicyclic graphs with maximum and minimum eccentric connectivity index subject to the certain graph constraints. Sharp lower and asymptotic upper bound for all graphs are given and various connections with other important graph invariants are established. In addition, we present explicit formulae for the values of eccentric connectivity index for several families of composite graphs and designed a linear algorithm for calculating the eccentric connectivity index of trees. Some open problems and related indices for further study are also listed.Comment: 25 pages, 5 figure

Laplacian spectral properties of signed circular caterpillars

A circular caterpillar of girth n is a graph such that the removal of all pendant vertices yields a cycle Cn of order n. A signed graph is a pair Γ = (G, σ), where G is a simple graph and σ ∶ E(G) → {+1, −1} is the sign function defined on the set E(G) of edges of G. The signed graph Γ is said to be balanced if the number of negatively signed edges in each cycle is even, and it is said to be unbalanced otherwise. We determine some bounds for the first n Laplacian eigenvalues of any signed circular caterpillar. As an application, we prove that each signed spiked triangle (G(3; p, q, r), σ), i. e. a signed circular caterpillar of girth 3 and degree sequence πp,q,r = (p + 2, q + 2, r + 2, 1,..., 1), is determined by its Laplacian spectrum up to switching isomorphism. Moreover, in the set of signed spiked triangles of order N, we identify the extremal graphs with respect to the Laplacian spectral radius and the first two Zagreb indices. It turns out that the unbalanced spiked triangle with degree sequence πN−3,0,0 and the balanced spike triangle (G(3; p, ^ q, ^ r^), +), where each pair in {p, ^ q, ^ r^} differs at most by 1, respectively maximizes and minimizes the Laplacian spectral radius and both the Zagreb indices

On the α\alpha-spectral radius of hypergraphs

For real $\alpha\in [0,1)$ and a hypergraph $G$, the $\alpha$-spectral radius of $G$ is the largest eigenvalue of the matrix $A_{\alpha}(G)=\alpha D(G)+(1-\alpha)A(G)$, where $A(G)$ is the adjacency matrix of $G$, which is a symmetric matrix with zero diagonal such that for distinct vertices $u,v$ of $G$, the $(u,v)$-entry of $A(G)$ is exactly the number of edges containing both $u$ and $v$, and $D(G)$ is the diagonal matrix of row sums of $A(G)$. We study the $\alpha$-spectral radius of a hypergraph that is uniform or not necessarily uniform. We propose some local grafting operations that increase or decrease the $\alpha$-spectral radius of a hypergraph. We determine the unique hypergraphs with maximum $\alpha$-spectral radius among $k$-uniform hypertrees, among $k$-uniform unicyclic hypergraphs, and among $k$-uniform hypergraphs with fixed number of pendant edges. We also determine the unique hypertrees with maximum $\alpha$-spectral radius among hypertrees with given number of vertices and edges, the unique hypertrees with the first three largest (two smallest, respectively) $\alpha$-spectral radii among hypertrees with given number of vertices, the unique hypertrees with minimum $\alpha$-spectral radius among the hypertrees that are not $2$-uniform, the unique hypergraphs with the first two largest (smallest, respectively) $\alpha$-spectral radii among unicyclic hypergraphs with given number of vertices, and the unique hypergraphs with maximum $\alpha$-spectral radius among hypergraphs with fixed number of pendant edges