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In this paper we investigate the hypergraphs whose spectral radii attain the maximum among all uniform hypergraphs with given number of edges. In particular we characterize the hypergraph(s) with maximum spectral radius over all unicyclic…

The eccentricity matrix of a connected graph $G$, denoted by $\mathcal{E}(G)$, is obtained from the distance matrix of $G$ by keeping the largest nonzero entries in each row and each column and leaving zeros in the remaining ones. The…

The $f$ adjacency matrix is a type of edge-weighted adjacency matrix, whose weight of an edge $ij$ is $f(d_i,d_j)$, where $f$ is a real symmetric function and $d_i,d_j$ are the degrees of vertex $i$ and vertex $j$. The $f$-spectral radius…

This paper presents a sharp upper bound for the spectral radius of simple digraphs with described number of arcs. Further, the extremal graphs which attain the maximum spectral radius among all simple digraphs with fixed arcs are…

We study the effect of three types of graft transformations to increase or decrease the distance spectral radius of uniform hypergraphs, and we determined the unique $k$-uniform hypertrees with maximum, second maximum, minimum and second…

The average distance of a vertex $v$ of a connected graph $G$ is the arithmetic mean of the distances from $v$ to all other vertices of $G$. The proximity $\pi(G)$ and the remoteness $\rho(G)$ of $G$ are the minimum and the maximum of the…

Bicyclic graph is a connected graph in which the number of edges equals the number of vertices plus one. In this paper, we determine the graph which alone maximizes the spectral radii among all the bicyclic graphs on $n$ vertices with fixed…

A spanning tree $T$ of a connected graph $G$ is a subgraph of $G$ that is a tree covers all vertices of $G$. The leaf distance of $T$ is defined as the minimum of distances between any two leaves of $T$. A fractional matching of a graph $G$…

Let $G$ be a simple finite connected graph with vertex set $V(G) = \{v_1,v_2,\ldots,v_n\}$. Denote the degree of vertex $v_i$ by $d_i$ for all $1 \leq i \leq n$. The Randi\'c matrix of $G$, denoted by $R(G) = [r_{i,j}]$, is the $n \times n$…

The spectral radius of a uniform hypergraph is defined to be that of the adjacency tensor the hypergraph. It is known that the unique unicyclic hypergraph with the largest spectral radius is a nonlinear hypergraph, and the unique linear…

We establish maximal trees and graphs for the difference of average distance and proximity proving thus the corresponding conjecture posed in [4]. We also establish maximal trees for the difference of average eccentricity and remoteness and…

The rank (also known as protection number or leaf-height) of a vertex in a rooted tree is the minimum distance between the vertex and any of its leaf descendants. We consider the sum of ranks over all vertices (known as the security) in…

There are several interrelated notions of discrete curvature on graphs. Many approaches utilize the optimal transportation metric on its probability simplex or the distance matrix of the graph. In this survey article, we compute formulas…

Suppose $G$ is a connected simple graph with the vertex set $V( G ) = \{ v_1,v_2,\cdots ,v_n \} $. Let $d_G( v_i,v_j ) $ be the least distance between $v_i$ and $v_j$ in $G$. Then the distance matrix of $G$ is $D( G ) =( d_{ij} ) _{n\times…

Let $G$ be an irregular graph on $n$ vertices with maximum degree $\Delta$ and diameter $D$. We show that \Delta-\lambda_1>\frac{1}{nD} where $\lambda_1$ is the largest eigenvalue of the adjacency matrix of $G$. We also study the effect of…

A design graph is a regular bipartite graph in which any two distinct vertices of the same part have the same number of common neighbors. This class of graphs have a close relationship to strongly regular graphs. In this paper, we study the…

For a simple, undirected and connected graph $G$, $D_{\alpha}(G) = \alpha Tr(G) + (1-\alpha) D(G)$ is called the $\alpha$-distance matrix of $G$, where $\alpha\in [0,1]$, $D(G)$ is the distance matrix of $G$, and $Tr(G)$ is the vertex…

We introduce the concept of distance ideals of graphs, which can be regarded as a generalization of the Smith normal form and the spectra of the distance matrix of a graph. We obtain a classification of the graphs with at most one trivial…

For a simple graph $G$, the $3$-distance graph, $D_3(G)$, is a graph with the vertex set $V(G)$ and two vertices are adjacent if and only if their distance is $3$ in the graph $G$. For a connected graph $G$, we provide some conditions for…

Let G be a graph of given order and mu(G) be the largest eigenvalue of its adjacency matrix. We give conditions on mu(G) that imply Hamiltonicity of G and of its complement.