Maximal reflexive cacti with four cycles: The approach via Smith graphs

• Published in: Linear algebra and its applications Vol. 435; no. 10; pp. 2530 - 2543
• Main Authors: RASAJSKI, M, RADOSAVLJEVIC, Z, MIHAILOVIC, B
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier Inc 15.11.2011; Elsevier
• Subjects: Algebra; Cactus; Combinatorics; Combinatorics. Ordered structures; Exact sciences and technology; Graph theory; Linear and multilinear algebra, matrix theory; Mathematics; Reflexive graph; Sciences and techniques of general use; Second largest eigenvalue; Smith graphs; Spectral graph theory; Smith graphs; Reflexive graph; 05C50; Second largest eigenvalue; Spectral graph theory; Cactus; Graph construction; Multigraph; Adjacency matrix; Graph theory; Hereditary; Spectral theory; Subgraph; Eigenvalue problem
• ISSN: 0024-3795; 1873-1856
• DOI: 10.1016/j.laa.2011.04.023

Cacti, or treelike graphs, are graphs whose all cycles are mutually edge-disjoint. Graphs with the property λ 2 ⩽ 2 are called reflexive graphs, where λ 2 is the second largest eigenvalue of the corresponding (0, 1)-adjacency matrix. The property λ 2 ⩽ 2 is a hereditary one, i.e. all induced subgraphs of a reflexive graph are also reflexive. This is why we represent reflexive graphs through the maximal graphs within a given class (such as connected cacti with a fixed number of cycles). In previous work we have determined all maximal reflexive cacti with four cycles whose cycles do not form a bundle and pointed out the role of so-called pouring of Smith graphs in their construction. In this paper, besides pouring, we show several other patterns of the appearance of Smith trees in those constructions. These include splitting of a Smith tree, adding an edge to a Smith tree and then splitting of the resulting graph, identifying two vertices of a Smith tree and then splitting the resulting graph. Our results show that the presence of Smith trees is evident in all such maximal reflexive cacti with four cycles and that in most of them Smith graphs appear in the described way.