相关论文: The dual Cheeger-Buser inequality for graphons
Whenever graphs admit equitable partitions, their quotient graphs highlight the structure evidenced by the partition. It is therefore very natural to ask what can be said about two graphs that have the same quotient according to certain…
Hladky, Hu, and Piguet [Tilings in graphons, preprint] introduced the notions of matching and fractional vertex covers in graphons. These are counterparts to the corresponding notions in finite graphs. Combinatorial optimization studies the…
The crossing number of a graph $G$ denotes the minimum number of crossings in any planar drawing of $G$. In this short note, we confirm a long-standing conjecture posed by Pach, Spencer, and T\'oth over 25 years ago, establishing an optimal…
The spectral theory of graphs provides a bridge between classical signal processing and the nascent field of graph signal processing. In this paper, a spectral graph analogy to Heisenberg's celebrated uncertainty principle is developed.…
We introduce a polynomial invariant of graphs on surfaces, $P_G$, generalizing the classical Tutte polynomial. Topological duality on surfaces gives rise to a natural duality result for $P_G$, analogous to the duality for the Tutte…
We consider the signless $p$-Laplacian of a graph, a generalisation of the usual signless Laplacian (the case $p=2$). We show a Perron-Frobenius property and basic inequalites for the largest eigenvalue and provide upper and lower bounds…
We prove a theorem that can be thought of as a common generalization of the Discrete Nodal Theorem and (one direction of) Cheeger's Inequality for graphs. special case of this result will assert that if the second and third eigenvalues of…
Spectral graph bisections are a popular heuristic aimed at approximating the solution of the NP-complete graph bisection problem. This technique, however, does not always provide a robust tool for graph partitioning. Using a special class…
We study a parameter of bipartite graphs called readability, introduced by Chikhi et al. (Discrete Applied Mathematics, 2016) and motivated by applications of overlap graphs in bioinformatics. The behavior of the parameter is poorly…
G. Cocco, E. Cadelano, and L. Colombo [Phys. Rev. B 81, 241412(R) (2010)] have suggested that combinations of shear and uniaxial strain can be used to open a band gap in graphene at much lower levels of strain than with the application of…
In [Steve Butler. A note about cospectral graphs for the adjacency and normalized Laplacian matrices. Linear Multilinear Algebra, 58(3-4):387-390, 2010.], Butler constructed a family of bipartite graphs, which are cospectral for both the…
In this paper, we give sufficient conditions on the spectral radius for a bipartite graph to Hamiltonian and traceable, which expand the results of Lu, Liu and Tian (2012) [10]. Furthermore, we also present tight sufficient conditions on…
Alon and Krivelevich conjectured that if $G$ is a bipartite graph of maximum degree $\Delta$, then the choosability (or list chromatic number) of $G$ satisfies $\chi_{\ell}(G) = O \left ( \log \Delta \right )$. Currently, the best known…
We present an algorithm for determining whether a bipartite graph $G$ is 2-chordal (formerly doubly chordal bipartite). At its core this algorithm is an extension of the existing efficient algorithm for determining whether a graph is…
We introduce a concept of isoperimetric dimension for magnetic graphs, that is, graphs where every edge is assigned a complex number of modulus one. In analogy with the classical case, we show that isoperimetric inequalities imply Sobolev…
In this paper, the dispersability of the Cartesian graph bundle over two cycles is completely solved. We show the Cartesian graph bundle $G$ over two cycles is dispersable if $G$ is bipartite; otherwise, $G$ is nearly dispersable.
The notion of travel groupoids was introduced by L. Nebesk\'y in 2006 in connection with a study on geodetic graphs. A travel groupoid is a pair of a set $V$ and a binary operation $*$ on $V$ satisfying two axioms. For a travel groupoid, we…
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…
In this article, we introduce a geometric and a spectral preorder relation on the class of weighted graphs with a magnetic potential. The first preorder is expressed through the existence of a graph homomorphism respecting the magnetic…
Intuitively speaking, a bipartite graph is mirror if it can be drawn in the Cartesian plane in such a way that, the vertices of one stable are points in x=0, the vertices of the other stable set are points in x=1, the edges are straight…