Some families of digraphs determined by the complementarity spectrum

Abstract

We examine the capacity of the complementarity spectrum to distinguish non-isomorphic digraphs. We focus on the seven families with exactly three complementarity eigenvalues. Our findings reveal that in some, but not all families, any two non-isomorphic members have different complementarity spectrum. Complementarity eigenvalues outperform traditional eigenvalues in the task of identifying graphs. Indeed, the question of whether graphs are uniquely determined by their complementarity spectrum remains unresolved, highlighting the significance of this tool in graph theory. Moreover, since the complementarity spectrum of a digraph was characterized as the set of spectral radii of the induced strongly connected subdigraphs, the results of this study provides useful structural information for important families of digraphs.

Figures (14)

• Figure 1: Digraph $\infty(3,5)$ and a schematic representation of the same digraph.
• Figure 2: Type 1 digraph (left) and Type 2 digraph (right)
• Figure 3: Digraph $\theta(0,2,1)$ and a schematic representation of the same digraph.
• Figure 4: Type 3 digraph (left), Type 4 digraph (center) and Type 5 digraph (right). Observe that one $\theta$-subdigraph can be identified in these three examples by taking the larger (round) cycle in each case, and any other arc.
• Figure 5: $\rho(\infty(5,5))<\rho(\infty(4,6))<\rho( \infty(3,7))<\rho( \infty(2,8))$.

Theorems & Definitions (39)

• Lemma 2.1
• Theorem 2.2
• Proposition 2.3
• Theorem 2.4
• Theorem 2.5
• Theorem 2.7
• Theorem 3.1
• Corollary 3.2
• Corollary 3.3
• Example 3.4