Related papers: Some extremal problems for hereditary properties o…
Let $G$ be a graph with $n$ vertices and $\lambda_n(G)$ be the least eigenvalue of its adjacency matrix of $G$. In this paper, we give sharp bounds on the least eigenvalue of graphs without given pathes or cycles and determine the extremal…
In this paper we deal with a Tur\'an-type problem: given a positive integer n and a forbidden graph H, how many edges can there be in a graph on n vertices without a subgraph H? How does a graph look like if it has this extremal edge…
We determine the extremal graph $G$ of order $n$ that maximizes the sum of the spectral radii of $G$ and its complement. This resolves a conjecture posed by Stevanovi\'{c} in 2007.
An almost self-centered graph is a connected graph of order $n$ with exactly $n-2$ central vertices, and an almost peripheral graph is a connected graph of order $n$ with exactly $n-1$ peripheral vertices. We determine (1) the maximum girth…
One of the earliest results in extremal graph theory, Mantel's theorem, states that the maximum number of edges in a triangle-free graph $G$ on $n$ vertices is $\lfloor n^2/4 \rfloor$. We investigate how this extremal bound is affected when…
Tur\'{a}n type extremal problem is how to maximize the number of edges over all graphs which do not contain fixed forbidden subgraphs. Similarly, spectral Tur\'{a}n type extremal problem is how to maximize (signless Laplacian) spectral…
Graph theory on surfaces extends classical graph structures to topological surfaces, providing a theoretical foundation for characterizing the embedding properties of complex networks in constrained spaces. The study of bounding the…
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…
We consider the problem of determining the inducibility (maximum possible asymptotic density of induced copies) of oriented graphs on four vertices. We provide exact values for more than half of the graphs, and very close lower and upper…
The classic extremal problem is that of computing the maximum number of edges in an $F$-free graph. In the case where $F=K_{r+1}$, the extremal number was determined by Tur\'an. Later results, known as supersaturation theorems, proved that…
Computer or communication networks are so designed that they do not easily get disrupted under external attack and, moreover, these are easily reconstructible if they do get disrupted. These desirable properties of networks can be measured…
Let F(G) be a fixed linear combination of the k extremal eigenvalues of a graph G and of its complement. The problem of finding max{F(G):v(G)=n} generalizes a number of problems raised previously in the literature. We show that the limit…
To compute the hyperbolicity constant is an almost intractable problem, thus it is natural to try to bound it in terms of some parameters of the graph. Let $\mathcal{G}(g,c,n)$ be the set of graphs $G$ with girth $g(G)=g$, circumference…
We establish tight lower and upper bounds on the number of edges in traceable graphs in several classes of dense graphs. A graph is traceable if it has a Hamiltonian path. We show that the bound is: - quadratic for the class of graphs of…
We give inequalities relating the eigenvalues of the adjacency matrix and the Laplacian of a graph, and its minimum and maximum degrees. The results are applied to derive new conditions for quasi-randomness of graphs.
We study the maximum number of edges in an $n$ vertex graph with Colin de Verdi\`{e}re parameter no more than $t$. We conjecture that for every integer $t$, if $G$ is a graph with at least $t$ vertices and Colin de Verdi\`{e}re parameter at…
A family of graphs is called degenerate if it contains at least one bipartite graph. In this paper, we investigate the spectral extremal problems for a degenerate family of graphs $\mathcal{F}$. By employing covering and independent…
We determine the maximum number of edges of an $n$-vertex graph $G$ with the property that none of its $r$-cliques intersects a fixed set $M\subset V(G)$. For $(r-1)|M|\ge n$, the $(r-1)$-partite Turan graph turns out to be the unique…
We prove properties of extremal graphs of girth 5 and order 20 <=v <= 32. In each case we identify the possible minimum and maximum degrees, and in some cases prove the existence of (non-trivial) embedded stars. These proofs allow for…
The $(k,r)$-fan is the graph consisting of $k$ copies of the complete graph $K_r$ which intersect in a single vertex, and is denoted by $F_{k,r}$. Erd\H{o}s, F\"uredi, Gould and Gunderson [J. Combin. Theory Ser. B 64 (1995) 89--100]…