Water is a basic natural resource for life and the sustainable development of society. Methods to assess water quality in freshwater ecosystems based on environmental quality bioindicators have proven to be low cost, reliable, and can be adapted to ecosystems with well-defined structures. The objective of this paper is to propose an interdisciplinary approach for the assessment of water quality in freshwater ecosystems through bioindicators. From the presence/absence of bioindicator organisms and their sensitivity/tolerance to environmental stress, we constructed a bipartite network, G. In this direction, we propose a new method that combines two research approaches, Graph Theory and Random Matrix Theory (RMT). Through the topological properties of the graph G, we introduce a topological index, called $JP(G)$ , to evaluate the water quality, and we study its properties and relationships with known indices, such as Biological Monitoring Working Party ( $BMWP$ ) and Shannon diversity ( $H^{\prime }$ ). Furthermore, we perform a scaling analysis of random bipartite networks with already specialized parameters for our case study. We validate our proposal for its application in the reservoir of Guájaro, Colombia. The results obtained allow us to infer that the proposed techniques are useful for the study of water quality, since they detect significant changes in the ecosystem.

We study the localization properties of the eigenvectors, characterized by their information entropy, of tight-binding random networks with balanced losses and gain. The random network model, which is based on Erdős–Rényi (ER) graphs, is defined by three parameters: the network size N, the network connectivity $\alpha$ , and the losses-and-gain strength $\gamma$ . Here, N and $\alpha$ are the standard parameters of ER graphs, while we introduce losses and gain by including complex self-loops on all vertices with the imaginary amplitude i $\gamma$ with random balanced signs, thus breaking the Hermiticity of the corresponding adjacency matrices and inducing complex spectra. By the use of extensive numerical simulations, we define a scaling parameter $\xi \equiv \xi (N,\alpha ,\gamma )$ that fixes the localization properties of the eigenvectors of our random network model; such that, when $\xi <0.1$ ( $10<\xi$ ), the eigenvectors are localized (extended), while the localization-to-delocalization transition occurs for $0.1<\xi <10$ . Moreover, to extend the applicability of our findings, we demonstrate that for fixed $\xi$ , the spectral properties (characterized by the position of the eigenvalues on the complex plane) of our network model are also universal; i.e., they do not depend on the specific values of the network parameters.

We investigate some topological and spectral properties of Erdős-Rényi (ER) random digraphs of size $n$ and connection probability $p, D(n,p)$. In terms of topological properties, our primary focus lies in analyzing the number of nonisolated vertices $V_x\left(D\right)$ as well as two vertex-degree-based topological indices: the Randić index $R\left(D\right)$ and sum-connectivity index $\chi \left(D\right)$. First, by performing a scaling analysis, we show that the average degree $\langle k\rangle$ serves as a scaling parameter for the average values of $V_x\left(D\right), R\left(D\right)$, and $\chi \left(D\right)$. Then, we also state expressions relating the number of arcs, largest eigenvalue, and closed walks of length 2 to $(n,p)$, the parameters of ER random digraphs. Concerning spectral properties, we observe that the eigenvalue distribution converges to a circle of radius $\sqrt{np(1-p)}$. Subsequently, we compute six different invariants related to the eigenvalues of $D(n,p)$ and observe that these quantities also scale with $\sqrt{np(1-p)}$. Additionally, we reformulate a set of bounds previously reported in the literature for these invariants as a function $(n,p)$. Finally, we phenomenologically state relations between invariants that allow us to extend previously known bounds.

We consider random geometric graphs on the plane characterized by a nonuniform density of vertices. In particular, we introduce a graph model where $n$ vertices are independently distributed in the unit disk with positions, in polar coordinates $(l,\theta )$, obeying the probability density functions $\rho \left(l\right)$ and $\rho \left(\theta \right)$. Here we choose $\rho \left(l\right)$ as a normal distribution with zero mean and variance $\sigma \in (0,\infty )$ and $\rho \left(\theta \right)$ as a uniform distribution in the interval $\theta \in [0,2\pi )$. Then, two vertices are connected by an edge if their Euclidean distance is less than or equal to the connection radius $\ell$. We characterize the topological properties of this random graph model, which depends on the parameter set $(n,\sigma ,\ell )$, by the use of the average degree $\langle k\rangle$ and the number of nonisolated vertices $V_{\times }$, while we approach their spectral properties with two measures on the graph adjacency matrix: the ratio of consecutive eigenvalue spacings $r$ and the Shannon entropy $S$ of eigenvectors. First we propose a heuristic expression for $\langle k(n,\sigma ,\ell )\rangle$. Then, we look for the scaling properties of the normalized average measure $\langle \overline{X}\rangle$ (where $X$ stands for $V_{\times }$, $r$, and $S$) over graph ensembles. We demonstrate that the scaling parameter of $\langle \overline{V_{\times }}\rangle =\langle V_{\times }\rangle /n$ is indeed $\langle k\rangle$, with $\langle \overline{V_{\times }}\rangle \approx 1-exp(-\langle k\rangle )$. Meanwhile, the scaling parameter of both $\langle \overline{r}\rangle$ and $\langle \overline{S}\rangle$ is proportional to $n^{-\gamma }\langle k\rangle$ with $\gamma \approx 0.16$.

We consider bipartite tight-binding graphs composed by N nodes split into two sets of equal size: one set containing nodes with on-site loss, the other set having nodes with on-site gain. The nodes are connected randomly with probability p. Specifically, we measure the connectivity between the two sets with the parameter α, which is the ratio of current adjacent pairs over the total number of possible adjacent pairs between the sets. For general undirected-graph setups, the non-Hermitian Hamiltonian H(γ,α,N) of this model presents pseudo-Hermiticity, where γ is the loss/gain strength. However, we show that for a given graph setup H(γ,α,N) becomes PT-symmetric. In both scenarios (pseudo-Hermiticity and PT-symmetric), depending on the parameter combination, the spectra of H(γ,α,N) can be real even when it is non-Hermitian. Then we demonstrate, for both setups, that there is a well-defined sector of the γα-plane (which grows with N) where the spectrum of H(γ,α,N) is predominantly real.

We perform a detailed statistical study of the distribution of topological and spectral indices on random graphs $G=(V,E)$ in a wide range of connectivity regimes. First, we consider degree-based topological indices (TIs), and focus on two classes of them: $X_Σ(G) = \sum_{uv \in E} f(d_u,d_v)$ and $X_Π(G) = \prod_{uv \in E} g(d_u,d_v)$, where $uv$ denotes the edge of $G$ connecting the vertices $u$ and $v$, $d_u$ is the degree of the vertex $u$, and $f(x,y)$ and $g(x,y)$ are functions of the vertex degrees. Specifically, we apply $X_Σ(G)$ and $X_Π(G)$ on Erdös-Rényi graphs and random geometric graphs along the full transition from almost isolated vertices to mostly connected graphs. While we verify that $P(X_Σ(G))$ converges to a standard normal distribution, we show that $P( X_Π(G))$ converges to a log-normal distribution. In addition we also analyze Revan-degree-based indices and spectral indices (those defined from the eigenvalues and eigenvectors of the graph adjacency matrix). Indeed, for Revan-degree indices, we obtain results equivalent to those for standard degree-based TIs. Instead, for spectral indices, we report two distinct patterns: the distribution of indices defined only from eigenvalues approaches a normal distribution, while the distribution of those indices involving both eigenvalues and eigenvectors approaches a log-normal distribution.