相关论文: On the Extremal Energy of Complex Unit Gain Dumbbe…

A complex unit gain graph ($ \mathbb{T} $-gain graph), $ \Phi=(G, \varphi) $ is a graph where the gain function $ \varphi $ assigns a unit complex number to each orientation of an edge of $ G $ and its inverse is assigned to the opposite…

We establish a lower bound for the energy of a complex unit gain graph in terms of the matching number of its underlying graph, and characterize all the complex unit gain graphs whose energy reaches this bound.

The sum of the absolute values of the eigenvalues of a graph is called the energy of the graph. We study the problem of finding graphs with extremal energy within specified classes of graphs. We develop tools for treating such problems and…

For a given simple graph $G$, the energy of $G$, denoted by $E(G)$, is defined as the sum of the absolute values of all eigenvalues of its adjacency matrix. Let $P_n^{\ell}$ be the unicyclic graph obtained by connecting a vertex of $C_\ell$…

We study Extremal Combinatorics problems where local properties are used to derive global properties. That is, we consider a given configuration where every small piece of the configuration satisfies some restriction, and use this local…

Let $G$ be a graph with $n$ non-isolated vertices and $m$ edges. The positive / negative square energies of $G$, denoted $s^+(G)$ / $s^-(G)$, are defined as the sum of squares of the positive / negative eigenvalues of the adjacency matrix…

The local properties problem of Erd\H{o}s and Shelah generalizes many Ramsey problems and some distinct distances problems. In this work, we derive a variety of new bounds for the local properties problem and its variants. We do this by…

We give a bound for the graph energy with given maximal degree in terms of the second and fourth moments of a graph. In the case in which the graph is $d$-regular we obtain the bound that is given in Van Dam, E. et al. (2014). through…

The energy of a simple graph $G$, denoted by $E(G)$, is defined as the sum of the absolute values of all eigenvalues of its adjacency matrix. Denote by $C_n$ the cycle, and $P_n^{6}$ the unicyclic graph obtained by connecting a vertex of…

The energy of a graph is the sum of the moduli of the eigenvalues of its adjacency matrix. We study the energy of integral circulant graphs, also called gcd graphs, which can be characterized by their vertex count $n$ and a set $\cal D$ of…

The binding number $b(G)$ of a graph, introduced by Woodall [J. Combin. Theory, Ser. B, 1973], is a central topic of both structural and extremal graph theory. It is closely related to fundamental combinatorial and structural properties of…

This paper surveys some recent results and progress on the extremal prob- lems in a given set consisting of all simple connected graphs with the same graphic degree sequence. In particular, we study and characterize the extremal graphs…

We study the energy per vertex in regular graphs. For every k, we give an upper bound for the energy per vertex of a k-regular graph, and show that a graph attains the upper bound if and only if it is the disjoint union of incidence graphs…

The eccentricity (anti-adjacency) matrix $\varepsilon(G)$ of a graph $G$ is obtained from the distance matrix by retaining the eccentricities in each row and each column. The $\varepsilon$-eigenvalues of a graph $G$ are those of its…

In this paper, we establish a couple of results on extremal problems in bipartite graphs. Firstly, we show that every sufficiently large bipartite graph with average degree $D$ and with $n$ vertices on each side has a balanced independent…

We study the uniqueness of optimal solutions to extremal graph theory problems. Lovasz conjectured that every finite feasible set of subgraph density constraints can be extended further by a finite set of density constraints so that the…

A graph $G$ is $[a,b]$-covered if for each edge $e$ of $G$ there is an $[a,b]$-factor containing it. For $a=b=1$, an $[a,b]$-covered graph is a matching covered graph. The structural theory of matching covered graphs constitutes a…

Let $G$ be a graph of order $n$, and let $A(G)$ and $D(G)$ be the adjacency matrix and the degree matrix of $G$ respectively. Define the convex linear combinations $A_\alpha (G)$ of $A (G)$ and $D (G) $ by $$A_\alpha (G)=\alpha…

In network flow problems, there is a well-known one-to-one relationship between extreme points of the feasibility region and trees in the associated undirected graph. The same is true for the dual differential problem. In this paper, we…

We systematically study a natural problem in extremal graph theory, to minimize the number of edges in a graph with a fixed number of vertices, subject to a certain local condition: each vertex must be in a copy of a fixed graph $H$. We…