An \((\alpha,\beta)\)-spanner of an \(n\)-vertex undirected, unweighted graph \(G=(V,E)\) is a subgraph \(H\) satisfying, for all \(u,v\in V\), \(\mathrm{dist}_H(u,v)\le \alpha\cdot \mathrm{dist}_G(u,v)+\beta\). For any \(k\in\mathbb{N}\), classical results show that a \((2k-1,0)\)-spanner with \(O(n^{1+1/k})\) edges exists and is asymptotically optimal under Erdős’ girth conjecture. This conditional lower bound applies only to adjacent pairs, leaving open the possibility of improved stretch for more distant pairs.

Spectral features are widely incorporated within Graph Neural Networks (GNNs) to improve their expressive power, or their ability to distinguish among non-isomorphic graphs. One popular example is the usage of graph Laplacian eigenvectors for positional encoding in MPNNs and Graph Transformers. The expressive power of such Spectrally-enhanced GNNs (SGNNs) is usually evaluated via the $k$-WL graph isomorphism test hierarchy and homomorphism counting. Yet, these frameworks align poorly with the graph spectra, yielding limited insight into SGNNs' expressive power. In this paper, we leverage a well-studied paradigm of classifying graphs by their largest eigenvalue multiplicity to introduce an expressivity hierarchy for SGNNs. We then prove that many SGNNs are incomplete even on graphs with distinct eigenvalues. To mitigate this deficiency, we adapt rotation equivariant neural networks to the graph spectra setting, yielding equiEPNN, a novel SGNN that provably improves upon contemporary SGNNs' expressivity on simple spectrum graphs. We then demonstrate that equiEPNN achieves perfect eigenvector canonicalization on ZINC, and performs favorably on image classification on MNIST-Superpixel and graph property regression on ZINC, compared to leading spectral methods.

In this paper, we focus on two related problems, the n-Pairs Shortest Paths (n-PSP) problem and the All-Nodes Shortest Cycles (ANSC) problem. In the n-PSP problem, given a graph G with n vertices and m edges, as well as a set P ⊆ V × V consisting of at most n pairs of vertices, our objective is to estimate the distances between each pair (u,v ) in P. In the ANSC problem, the objective is to find for each node the shortest cycle that includes that particular node. In both problems, we present new algorithms and reductions that enhance the existing solutions in terms of both time complexity and approximation factor.

In this paper we provide a Õ(n2) time algorithm that computes a 2-multiplicative approximation of the girth of an n-node m-edge directed graph with non-negative edge weights. We also provide an additional algorithm that computes a 2-multiplicative approximation of the girth in time 1. Our results naturally provide algorithms for improved constructions of 4-roundtrip spanners, the analog of spanners in directed graphs. Our algorithm is optimal (up to a log n factor) for dense graphs with m = Θ(n2). For comparison, previously, the best approximation ratio with a similar running time for dense graphs was O(log n log log n) [1]. Moreover, unlike previous algorithms, our algorithm neither assumes integer weights, nor does it depend on the maximum edge weight of the graph.