The Aα-Spectral Radius and k-Extendability in Graphs

In Math Program 55(1992), 129–168, Conforti and Rao conjectured that every balanced bipartite graph contains an edge that is not the unique chord of a cycle. We prove this conjecture for balanced bipartite graphs that do not contain a cycle of length 4 (also known as linear balanced bipartite graphs), and for balanced bipartite graphs whose maximum degree is at most 3. We in fact obtain results for more general classes, namely linear balanceable and subcubic balanceable graphs. Additionally, we prove that cubic balanced graphs contain a pair of twins, a result that was conjectured by Morris, Spiga, and Webb in ( Discrete Math 310(2010), 3228–3235).

On the (signless Laplacian) spectral radius of minimally [formula omitted]-(edge)-connected graphs for small [formula omitted

Let G be a connected graph with distance matrix , and let be the diagonal matrix of vertex transmissions of G. For any , the -matrix of G is defined as In this paper, we study the -spectra of graphs. Firstly, the -eigenvalues of some special graphs are presented. Then we give a lower bound on the kth smallest -eigenvalue of graphs, and the extremal graphs are characterized. Also, several graph transformations on the -spectral radius are given, as applications, some extremal graphs with given structure parameters are characterized. Finally, we give some properties when two graphs have the same -spectra and several graphs are proved to be determined by their -spectra.

Some graft transformations and its applications on the distance spectral radius of a graph

On the sum of all distances in bipartite graphs

For $ \alpha \in [0,1]$, let $A_{\alpha}(G) = \alpha D(G) +(1-\alpha)A(G)$ be $A_{\alpha}$-matrix, where $A(G)$ is the adjacent matrix and $D(G)$ is the diagonal matrix of the degrees of a graph $G$. Clearly, $A_{0} (G)$ is the adjacent matrix and $2 A_{\frac{1}{2}}$ is the signless Laplacian matrix. A connected graph is a cactus graph if any two cycles of $G$ have at most one common vertex. We first propose the result for subdivision graphs, and determine the cacti maximizing $A_{\alpha}$-spectral radius subject to fixed pendant vertices. In addition, the corresponding extremal graphs are provided. As consequences, we determine the graph with the $A_{\alpha}$-spectral radius among all the cacti with $n$ vertices; we also characterize the $n$-vertex cacti with a perfect matching having the largest $A_{\alpha}$-spectral radius.

The Tur\'an type extremal problem asks to maximize the number of edges over all graphs which do not contain fixed subgraphs. Similarly, the spectral Tur\'an type extremal problem asks to maximize spectral radius of all graphs which do not contain fixed subgraphs. In this paper, we determine the maximum spectral radius of all graphs without containing a linear forest as a subgraph and characterize all corresponding extremal graphs. In addition, the maximum number of edges and spectral radius of all bipartite graphs without containing $k\cdot P_3$ as a subgraph are obtained and all extremal graphs are also characterized. Moreover, some relations between Tu\'an type extremal problems and spectral Tur\'an type extremal problems are discussed.

Let $A(G)$ and $D(G)$ be the adjacency matrix and the degree diagonal matrix of a graph $G$, respectively. Then $L(G)=D(G)-A(G)$ is called Laplacian matrix of the graph $G$. Let $G$ be a graph with $n$ vertices and $m$ edges. Then the $LI$-matrix of $G$ is defined as $LI(G)=L(G)-\frac{2m}{n}I_n$, where $I_n$ is the identity matrix. In this paper, we are interested in extremal properties of the Ky Fan $k$-norm of the $LI$-matrix of graphs, which is closely related to the well known problems and results in spectral graph theory, such as the Laplacian spectral radius, the Laplacian spread, the sum of the $k$ largest Laplacian eigenvalues, the Laplacian energy, and other parameters. Some bounds on the Ky Fan $k$-norm of the $LI$-matrix of graphs are given, and the extremal graphs are partly characterized. In addition, upper and lower bounds on the Ky Fan $k$-norm of $LI$-matrix of trees, unicyclic graphs, and bicyclic graphs are determined, and the corresponding extremal graphs are characterized.

Let $G=(V,\,E)$ be a simple graph of order $n$ and the normalized Laplacian eigenvalues $\rho_1\geq \rho_2\geq \cdots\geq\rho_{n-1}\geq \rho_n=0$. The normalized Laplacian energy (or Randi\'c energy) of $G$ without any isolated vertex is defined as $$RE(G)=\sum_{i=1}^{n}|\rho_i-1|.$$ In this paper, a lower bound on $\rho_1$ of connected graph $G$ ($G$ is not isomorphic to complete graph) is given and the extremal graphs (that is, the second minimal normalized Laplacian spectral radius of connected graphs) are characterized. Moreover, Nordhaus-Gaddum type results for $\rho_1$ are obtained. Recently, Gutman et al.~gave a conjecture on Randi\'c energy of connected graph [I. Gutman, B. Furtula, \c{S}. B. Bozkurt, On Randi\'c energy, Linear Algebra Appl. 442 (2014) 50--57]. Here this conjecture for starlike trees is proven.

A graph property is monotone if it is closed under removal of vertices and edges. In this paper we consider the following edge-deletion problem; given a monotone property P and a graph G, compute the smallest number of edge deletions that are needed in order to turn G into a graph satisfying P. We denote this quantity by E/sub P/'(G). The first result of this paper states that the edge-deletion problem can be efficiently approximated for any monotone property. 1) For any /spl epsiv/ > 0 and any monotone property P, there is a deterministic algorithm, which given a graph G of size n, approximates E/sub P/'(G) in time O(n/sup 2/) to within an additive error of /spl epsiv/n/sup 2/. Given the above, a natural question is for which monotone properties one can obtain better additive approximations of E/sub P/'. Our second main result essentially resolves this problem by giving a precise characterization of the monotone graph properties for which such approximations exist; 1. If there is a bipartite graph that does not satisfy P, then there is a /spl delta/ > 0 for which it is possible to approximate E/sub P/' to within an additive error of n/sup 2-/spl delta// in polynomial time. 2) On the other hand, if all bipartite graphs satisfy P, then for any /spl delta/ > 0 it is NP-hard to approximate E/sub P/' to within an additive error of n/sup 2-/spl delta//. While the proof of (1) is simple, the proof of (2) requires several new ideas and involves tools from extremal graph theory together with spectral techniques. This approach may be useful for obtaining other hardness of approximation results. Interestingly, prior to this work it was not even known that computing E/sub P/' precisely for the properties in (2) is NP-hard. We thus answer (in a strong form) a question of Yannakakis [1981], who asked in 1981 if it is possible to find a large and natural family of graph properties for which computing E/sub P/' is NP-hard.

New explicit constructions of infinite families of finite small world graphs of large girth with well-defined projective limits which is an infinite tree are described. The applications of these objects to constructions of LDPC codes and cryptographic algorithms are shortly observed. We define families of homogeneous algebraic graphs of large girth over the commutative ring K. For each commutative integrity ring K with |K| > 2, we introduce a family of bipartite homogeneous algebraic graphs of large girth over K formed by graphs with sets of points and lines isomorphic to Kn, n > 1, and cycle indicator ≥ 2n + 2 such that their projective limit is well defined and isomorphic to an infinite forest.

Suppose π = (d_1,d_2,...,d_n) and π′ = (d′_1,d′_2,...,d′_n) are two positive non- increasing degree sequences, write π ⊳ π′ if and only if π \neq π′, \sum_{i=1}^n d_i = \sum_{i=1}^n d′_i, and \sum_{i=1}^j d_i ≤ \sum_{i=1}^j d′_i for all j = 1, 2, . . . , n. Let ρ(G) and μ(G) be the spectral radius and signless Laplacian spectral radius of G, respectively. Also let G and G′ be the extremal graphs with the maximal (signless Laplacian) spectral radii in the class of connected graphs with π and π′ as their degree sequences, respectively. If π ⊳ π′ can deduce that ρ(G) < ρ(G′) (respectively, μ(G) < μ(G′)), then it is said that the spectral radii (respectively, signless Laplacian spectral radii) of G and G′ satisfy the majorization theorem. This paper presents a survey to the recent results on the theory and application of the majorization theorem in graph spectrum and topological index theory.

A graph is common if the number of monochromatic copies of in a 2‐edge‐coloring of the complete graph is asymptotically minimized by the random coloring. The classification of common graphs is one of the most intriguing problems in extremal graph theory. We study the notion of weakly locally common graphs considered by Csóka, Hubai, and Lovász [arXiv:1912.02926], where the graph is required to be the minimizer with respect to perturbations of the random 2‐edge‐coloring. We give a complete analysis of the 12 initial terms in the Taylor series determining the number of monochromatic copies of in such perturbations and classify graphs based on this analysis into three categories:Graphs of Class I are weakly locally common. Graphs of Class II are not weakly locally common. Graphs of Class III cannot be determined to be weakly locally common or not based on the initial 12 terms. As a corollary, we obtain new necessary conditions on a graph to be common and new sufficient conditions on a graph to be not common.

On the sum of the squares of all distances in bipartite graphs with given connectivity

We prove Ramsey-type results for intersection graphs of geometric objects. In particular, we prove the following bounds, all of which are tight apart from the constant c. There is a constant c > 0 such that for every family F of n convex sets in the plane, the intersection graph of F or its complement contains a balanced complete bipartite graph of size at least cn. There is a constant c > 0 such that for every family F of n x-monotone curves in the plane, the intersection graph G of F contains a balanced complete bipartite graph of size at least cn/log n or the complement of G contains a balanced complete bipartite graph of size at least cn. Our bounds rely on new Turan-type results on incomparability graphs of partially ordered sets.