Extremal distance spectral radius of graphs with fixed size

A fractional matching of $G$ is a function $f: E(G)\to [0,1]$ such that $\sum_{e\in E_G(v_i)}f(e)\le 1$ for any $v_i\in V(G)$, where $E_G(v_i)=\{e: e\in E(G) \ \textrm{and}\ e \ \textrm{is incident with} \ v_i\}$. Let $\alpha_f(G)$ denote the fractional matching number of $G$, which is defined as $\alpha_f(G)=\max\{\sum_{e\in E(G)}f(e): f\ \textrm{is a fractional matching of} \ G\}$. Let $\{G_1,G_2,G_3,\dots\}$ be a set of graphs, a $\{G_1,G_2,G_3,\dots\}$-factor of a graph $G$ is a spanning subgraph of $G$ such that each component of which is isomorphic to one of $\{G_1,G_2,G_3,\dots\}$. In this paper, we first establish a sharp upper bound for the distance spectral radius to guarantee that $\alpha_f(G)>\frac{n-k}{2}$ in a graph $G$ of order $n$ with given minimum degree, where $0<k<n$ is an integer. Then we give a sharp upper bound on the distance spectral radius of a graph $G$ with given minimum degree $\delta$ to ensure that $G$ has a $\{K_2, \{C_k\}\}$-factor, where $3\le k<+\infty$ is an integer. Moreover, we obtain a sharp upper bound on the distance spectral radius for the existence of a $\{K_{1,1},K_{1,2},\dots,K_{1,k}\}$-factor with $2\le k<+\infty$ in a graph $G$ with given minimum degree.

Connectivity, diameter, independence number and the distance spectral radius of graphs

Some graft transformations and its applications on the distance spectral radius of a graph

Let D(G) denote the distance matrix of a connected graph G. The largest eigenvalue of D(G) is called the distance spectral radius of a graph G, denoted by ϱ(G). In this article, we give sharp upper and lower bounds for the distance spectral radius and characterize those graphs for which these bounds are best possible.

On distance spectral radius of graphs

Some graft transformations and its application on a distance spectrum

The graphs with smallest, respectively largest, distance spectral radius among the connected graphs, respectively trees with a given number of odd vertices, are determined. Also, the graphs with the largest distance spectral radius among the trees with a given number of vertices of degree 3, respectively given number of vertices of degree at least 3, are determined. Finally, the graphs with the second and third largest distance spectral radius among the trees with all odd vertices are determined.

Let D(G) be the distance matrix of a connected graph G. The distance spectral radius of G is the largest eigenvalue of D(G) and it has been proposed to be a molecular structure descriptor. In this article, we determine the unique trees with minimal and maximal distance spectral radii among trees with fixed bipartition. As a corollary, the trees with the first three minimal distance spectral radii are determined. Furthermore, we determine the unique trees with minimal distance spectral radii among n-vertex trees with fixed number of pendent vertices or fixed even diameter, respectively. We also propose a conjecture regarding the tree with minimal distance spectral radius among n-vertex trees with fixed odd diameter.

In this article, we study how the distance spectral radius behaves when the graph is perturbed by grafting edges. As applications, we also determine the graph with k cut vertices (respectively, k cut edges) with the minimal distance spectral radius.

Characterizing star factors via the size, the spectral radius or the distance spectral radius of graphs

Sharp bounds on distance spectral radius of graphs

This paper studies some graph distance spectrum, first introduces the basic conce pts that used in this article, and related terms and the main results obtained mark in this p aper. At the same time the paper depicts the matching number is β, order number is n that has minimum distance spectral radius of graph in all connected graph. The article finally c alculated the distance spectrum of G⊙K2 and 123 GG G  .

On distance spectral radius of graphs with given number of pendant paths of fixed length

Distance spectral radius of trees with given matching number

The distance spectral radius of a connected hypergraph is the largest eigenvalue of its distance matrix. In this paper we present a new transformation that decreases distance spectral radius. As applications, if ? ? ?m+1 2?, we determine the unique k-uniform hypertree of fixed m edges and maximum degree ? with the minimumdistance spectral radius. And we characterize the k-uniform hypertrees on m edges with the fourth, fifth, and sixth smallest distance spectral radius. In addition, we obtain the k-uniform hypertree on m edges with the third largest distance spectral radius.