Efficient Neural Network Verification via Layer-based Semidefinite Relaxations and Linear Cuts

We introduce an efficient and tight layer-based semidefinite relaxation for verifying local robustness of neural networks. The improved tightness is the result of the combination between semidefinite relaxations and linear cuts. We obtain a computationally efficient method by decomposing the semidefinite formulation into layerwise constraints. By leveraging on chordal graph decompositions, we show that the formulation here presented is provably tighter than current approaches. Experiments on a set of benchmark networks show that the approach here proposed enables the verification of more instances compared to other relaxation methods. The results also demonstrate that the SDP relaxation here proposed is one order of magnitude faster than previous SDP methods.

On the Eccentric Connectivity Polynomial of ℱ-Sum of Connected Graphs

The eccentric connectivity polynomial (ECP) of a connected graph G=VG,EG is described as ξcG,y=∑a∈VGdegGayecGa, where ecGa and degGa represent the eccentricity and the degree of the vertex a, respectively. The eccentric connectivity index (ECI) can also be acquired from ξcG,y by taking its first derivatives at y=1. The ECI has been widely used for analyzing both the boiling point and melting point for chemical compounds and medicinal drugs in QSPR/QSAR studies. As the extension of ECI, the ECP also performs a pivotal role in pharmaceutical science and chemical engineering. Graph products conveniently play an important role in many combinatorial applications, graph decompositions, pure mathematics, and applied mathematics. In this article, we work out the ECP of ℱ-sum of graphs. Moreover, we derive the explicit expressions of ECP for well-known graph products such as generalized hierarchical, cluster, and corona products of graphs. We also apply these outcomes to deduce the ECP of some classes of chemical graphs.

A rainbow blow-up lemma for almost optimally bounded edge-colourings

A subgraph of an edge-coloured graph is called rainbow if all its edges have different colours. We prove a rainbow version of the blow-up lemma of Komlós, Sárközy, and Szemerédi that applies to almost optimally bounded colourings. A corollary of this is that there exists a rainbow copy of any bounded-degree spanning subgraph H in a quasirandom host graph G, assuming that the edge-colouring of G fulfills a boundedness condition that is asymptotically best possible. This has many applications beyond rainbow colourings: for example, to graph decompositions, orthogonal double covers, and graph labellings.

Some Results on Rosa-type Labelings of Graphs

Labelings that are used in graph decompositions are called Rosa-type labelings. The gamma-labeling of an almost-bipartite graph is a natural generalization of an alpha-labeling of a bipartite graph. It is known that if a bipartite graph G with m edges possesses an alpha-labeling or an almost-bipartite graph G with m edges possesses a gamma-labeling, then the complete graph K_{2mx+1} admits a cyclic G-decomposition. A variation of an alpha-labeling is introduced in this paper by allowing additional vertex labels and some conditions on edge labels and show that whenever a bipartite graph G admits such a labeling, then there exists a supergraph H of G such that H is almost-bipartite and H has a gamma-labeling.