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The Turan problem asks for the largest number of edges in an $n$-vertex graph not containing a fixed forbidden subgraph $F$. We construct a new family of graphs not containing $K_{s,t}$, for $t= C^s$, with $Omega(n^{2-1/s})$ edges matching the upper bound of Kovari, Sos and Turan.
The Laplacian spread of a graph is the difference between the largest eigenvalue and the second-smallest eigenvalue of the Laplacian matrix of the graph. We find that the class of strongly regular graphs attains the maximum of largest eigenvalues, th
Given a graph $H$, a graph is $H$-free if it does not contain $H$ as a subgraph. We continue to study the topic of extremal planar graphs, that is, how many edges can an $H$-free planar graph on $n$ vertices have? We define $ex_{_mathcal{P}}(n,H)$ to
An edge-coloring of a connected graph $G$ is called a {em monochromatic connection coloring} (MC-coloring for short) if any two vertices of $G$ are connected by a monochromatic path in $G$. For a connected graph $G$, the {em monochromatic connection
The first-fit coloring is a heuristic that assigns to each vertex, arriving in a specified order $sigma$, the smallest available color. The problem Grundy Coloring asks how many colors are needed for the most adversarial vertex ordering $sigma$, i.e.
An extension of the well-known Szeged index was introduced recently, named as weighted Szeged index ($textrm{sz}(G)$). This paper is devoted to characterizing the extremal trees and graphs of this new topological invariant. In particular, we proved t