相关论文: Extremal norms of graphs and matrices

Two-sample tests utilizing a similarity graph on observations are useful for high-dimensional and non-Euclidean data due to their flexibility and good performance under a wide range of alternatives. Existing works mainly focused on sparse…

This paper derives exponential tail bounds and polynomial moment inequalities for the spectral norm deviation of a random matrix from its mean value. The argument depends on a matrix extension of Stein's method of exchangeable pairs for…

The $t$-e.c. and pseudo-random property are typical properties of random graphs. In this note, we study the gap between them which has not been studied well. As a main result, we give the first explicit construction of infinite families of…

The matching energy of a graph was introduced by Gutman and Wagner, which is defined as the sum of the absolute values of the roots of the matching polynomial of the graph. For the random graph $G_{n,p}$ of order $n$ with fixed probability…

Several new trace norm inequalities are established for 2n x 2n block matrices, in the special case where the four n x n blocks are diagonal. Some of the inequalities are non-commutative analogs of Hanner's inequality, others describe the…

The eccentricity matrix of a graph is obtained from the distance matrix by keeping the largest entries in their row or column, and the remaining entries are replaced by zeros. The eccentricity energy of a graph is the sum of the absolute…

Under the unifying umbrella of a general result of Penrose & Yukich [Ann. Appl. Probab., (2003) 13, 277--303] we give laws of large numbers (in the $L^p$ sense) for the total power-weighted length of several nearest-neighbour type graphs on…

In this note, we investigate fundamental relations between exploration processes in random graphs, and branching processes. We formulate a class of models that we call {\em rank-$k$ random graphs}, and that are special in that their…

The feasibility conditions obtained in a previous report are refined, and used to determine several infinite families of feasible parameters for strongly regular graphs with no triangles. The methods are also used to improve the lower bound…

An extremal graph for a graph $H$ on $n$ vertices is a graph on $n$ vertices with maximum number of edges that does not contain $H$ as a subgraph. Let $T_{n,r}$ be the Tur\'{a}n graph, which is the complete $r$-partite graph on $n$ vertices…

In a graph, nodes can be characterized locally (with their degree $k$) or globally (e.g. with their average length path $\xi$ to other nodes). Here we investigate how $\xi$ depends on $k$. Our earlier algorithm of the construction of the…

Given a planar graph family $\mathcal{F}$, let ${\rm ex}_{\mathcal{P}}(n,\mathcal{F})$ and ${\rm spex}_{\mathcal{P}}(n,\mathcal{F})$ be the maximum size and maximum spectral radius over all $n$-vertex $\mathcal{F}$-free planar graphs,…

The definition of edge-regularity in graphs is a relaxation of the definition of strong regularity, so strongly regular graphs are edge-regular and, not surprisingly, the family of edge-regular graphs is much larger and more diverse than…

For a graph $H$, the {\em extremal number} $ex(n,H)$ is the maximum number of edges in a graph of order $n$ not containing a subgraph isomorphic to $H$. Let $\delta(H)>0$ and $\Delta(H)$ denote the minimum degree and maximum degree of $H$,…

The modeling of diffusion processes on graphs is the basis for many network science and machine learning approaches. Entropic measures of network-based diffusion have recently been employed to investigate the reversibility of these…

We investigate spectral properties of quantum graphs in the form of a periodic chain of rings with a connecting link between each adjacent pair, assuming that wave functions at the vertices are matched through conditions manifestly…

In this paper we are concerned with various graph invariants (girth, diameter, expansion constants, eigenvalues of the Laplacian, tree number) and their analogs for weighted graphs -- weighing the graph changes a combinatorial problem to…

A graph homomorphism from the rooted $d$-branching tree $\phi: T^d \to H$ is said to be cold if the values of $\phi$ for vertices arbitrarily far away from the root can restrict the value of $\phi$ at the root. Warmth is a graph parameter…

We obtain several sharp spectral bounds, approximations, and exact values for the isoperimetric number and related edge-expansion parameters of graphs. Our results focus on graph powers and on families of graphs with rich algebraic or…

A $k$-leaf power of a tree $T$ is a graph $G$ whose vertices are the leaves of $T$ and whose edges connect pairs of leaves whose distance in $T$ is at most $k$. A graph is a leaf power if it is a $k$-leaf power for some $k$. Over 20 years…