Qualitative, statistical and extreme properties of spectral indices of signable pseudo-invertible graphs

Abstract

In this paper, we investigate the Moore-Penrose inversion of a simple connected graph. We analyze qualitative, statistical, and extreme properties of spectral indices of signable pseudo-invertible graphs. We introduce and analyze a wide class of signable pseudo-invertible simple connected graphs. It is a generalization of the classical concept of positively integrally invertible graphs due to Godsil. We present several constructions of signable pseudo-invertible graphs. We also discuss statistical properties of various spectral indices of the class of signable pseudo-invertible graphs.

Figures (10)

• Figure 1: Top row: examples of a positively (left), negatively (middle), positively and negatively (right) pseudo-invertible graphs on $m=5$ vertices. Bottom row: corresponding weighted pseudo-inverse graphs.
• Figure 2: The uncomplete positively and negatively psudoinvertible bipartite graph $K_{m,m}^{-e}$ with $m=4$ (left) and its pseudo-inverse weighted graph $(K_{m,m}^{-e})^\dagger$ (right).
• Figure 3: A positively integrally invertible vertex labeled graph $G^{\mathscr{A}}$ with $k=4$ vertices (left); the $G^\mathscr{A}$ complete graph $G^{\mathscr{A}}_{1,2,1,1}$ complete graph (middle); the weighted signable pseudo-inverse graph $(G^{\mathscr{A}}_{1,2,1,1})^\dagger$ (right).
• Figure 4: Left: the number of all simple connected graphs of order $m\le 10$ (blue). The numbers of positively but not negatively pseudo-invertible graphs (magenta), the number of negatively but not positively pseudo-invertible graphs (red), the number of positively and negatively (bipartite) pseudo-invertible graphs (green). Right: computational time complexity of the results summarized in Table \ref{['tab-1']}.
• Figure 5: Signable pseudo-invertible graphs on $3\le m \le10$ vertices with a maximal value of $\lambda_{max}$ (see Table \ref{['tab-spektrumpositivepseudo']}).

Theorems & Definitions (22)

• Definition 1
• Definition 2
• Proposition 3
• Proof 1
• Proposition 4
• Proof 2
• Proposition 5
• Proof 3
• Proposition 6
• Proof 4