Spectral Threshold for Extremal Cyclic Edge-Connectivity

• Published in: Graphs and combinatorics Vol. 37; no. 6; pp. 2079 - 2093
• Main Authors: Aksoy, Sinan G., Kempton, Mark, Young, Stephen J.
• Format: Journal Article
• Language: English
• Published: Tokyo Springer Japan 01.11.2021; Springer Nature B.V; Springer
• Subjects: Combinatorics; Connectivity; Eigenvalues; Engineering Design; fault tolerance; graph analysis; Graph theory; Mathematics; MATHEMATICS AND COMPUTING; Mathematics and Statistics; Original Paper; spectral graph theory; Upper bounds; Graph spectrum; Cyclic edge connectivity; Edge connectivity
• ISSN: 0911-0119; 1435-5914
• DOI: 10.1007/s00373-021-02333-6

The universal cyclic edge-connectivity of a graph G is the least k such that there exists a set of k edges whose removal disconnects G into components where every component contains a cycle. We show that for graphs of minimum degree at least 3 and girth g at least 4, the universal cyclic edge-connectivity is bounded above by ( Δ - 2 ) g where Δ is the maximum degree. We then prove that if the second eigenvalue of the adjacency matrix of a d -regular graph of girth g ≥ 4 is sufficiently small, then the universal cyclic edge-connectivity is ( d - 2 ) g , providing a spectral condition for when this upper bound on universal cyclic edge-connectivity is tight.