Related papers: An Optimization Approach to Degree Deviation and S…

Let $G$ be a digraph with adjacency matrix $A(G)$. Let $D(G)$ be the diagonal matrix with outdegrees of vertices of $G$. Nikiforov \cite{Niki} proposed to study the convex combinations of the adjacency matrix and diagonal matrix of the…

For a connected graph $G$ with order $n$ and an integer $k\geq 1$, we denote by $$S_k(D(G))=\lambda_1(D(G))+\cdots+\lambda_k(D(G))$$ the sum of $k$ largest distance eigenvalues of $G$. In this paper, we consider the sharp upper bound and…

Let $G$ be a random graph on the vertex set $\{1,2,..., n\}$ such that edges in $G$ are determined by independent random indicator variables, while the probability $p_{ij}$ for $\{i,j\}$ being an edge in $G$ is not assumed to be equal.…

The degree/diameter problem is the problem of finding the largest possible number of vertices $n_{\Delta,D}$ in a graph of given degree $\Delta$ and diameter $D$. We consider the problem for the case of diameter $D=2$. William G Brown gave…

Let $G$ be an $n$-vertex graph, and let $\lambda(G)$ and $\lambda_n(G)$ denote the largest and smallest eigenvalues of its adjacency matrix. Write $e(G)$ for the number of edges of $G$, $d(G)=2e(G)/n$ for its average degree, and $T_r(n)$…

We show that for any constant $\Delta \ge 2$, there exists a graph $G$ with $O(n^{\Delta / 2})$ vertices which contains every $n$-vertex graph with maximum degree $\Delta$ as an induced subgraph. For odd $\Delta$ this significantly improves…

Let G be a simple connected graph of order n with degree sequence d_1, d_2, ..., d_n in non-increasing order. The spectral radius rho(G) of G is the largest eigenvalue of its adjacency matrix. For each positive integer L at most n, we give…

The spread of a real symmetric matrix is defined as the difference between its largest and smallest eigenvalue. The study of graph-related matrices has attracted considerable attention, leading to a substantial body of findings. In this…

We prove that every $n$-vertex $K_t$-minor-free graph $G$ of maximum degree $\Delta$ has a set $F$ of $O(t^2(\log t)^{1/4}\sqrt{\Delta n})$ edges such that every component of $G - F$ has at most $n/2$ vertices. This is best possible up to…

We present a new method for upper bounding the second eigenvalue of the Laplacian of graphs. Our approach uses multi-commodity flows to deform the geometry of the graph; we embed the resulting metric into Euclidean space to recover a bound…

Graph representations are the generalization of geometric graph drawings from the plane to higher dimensions. A method introduced by Tutte to optimize properties of graph drawings is to minimize their energy. We explore this minimization…

Let $\mathscr{G}_{n,\beta}$ be the set of graphs of order $n$ with given matching number $\beta$. Let $D(G)$ be the diagonal matrix of the degrees of the graph $G$ and $A(G)$ be the adjacency matrix of the graph $G$. The largest eigenvalue…

We show that for a graph $G$ with the vertex set $V$ and the largest eigenvalue $\lambda_{\max}(G)$, letting $$ M(G) := \max_{X,Y \subset V} \frac{e(X,Y)}{\sqrt{|X||Y|}} $$ (where $e(X,Y)$ denotes the number of edges between $X$ and $Y$),…

We study the classical optimization problem $\min_{x \in \mathbb{R}^d} f(x)$ and analyze the gradient descent (GD) method in both nonconvex and convex settings. It is well-known that, under the $L$-smoothness assumption ($\|\nabla^2 f(x)\|…

The problem of determining the optimal minimum degree condition for a balanced bipartite graph on 2ms vertices to contain m vertex disjoint copies of K_{s,s} was solved by Zhao. Later Hladk\'y and Schacht, and Czygrinow and DeBiasio…

Given a graph $G$, we would like to find (if it exists) the largest induced subgraph $H$ in which there are at least $k$ vertices realizing the maximum degree of $H$. This problem was first posed by Caro and Yuster. They proved, for…

Let $G$ be a connected nonregular graphs of order $n$ with maximum degree $\Delta$ that attains the maximum spectral radius. Liu and Li (2008) proposed a conjecture stating that $G$ has a degree sequence $(\Delta,\ldots,\Delta,\delta)$ with…

In this study we are interested mainly in investigating the relations between two graph irregularity measures which are widely used for structural irregularity characterization of connected graphs. Our study is focused on the comparison and…

Let $G$ be a graph with adjacency matrix $A(G)$ and let $D(G)$ be the diagonal matrix of the degrees of $G$. For any real $\alpha\in [0,1]$, Nikiforov [Merging the $A$- and $Q$-spectral theories, Appl. Anal. Discrete Math. 11 (2017)…

Let $G$ be a simple graph, and denote by $\lambda(G)$ its spectral radius. Sun and Das (2020) established that for any non-isolated vertex $v$ with degree $d(v)$, \[ \lambda(G)\leq \sqrt{\lambda(G-v)^2 + 2d(v) - 1}, \] which is a conjecture…