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We investigate some topological and spectral properties of Erd\H{o}s-R\'{e}nyi (ER) random digraphs $D(n,p)$. In terms of topological properties, our primary focus lies in analyzing the number of non-isolated vertices $V_x(D)$ as well as two vertex-degree-based topological indices: the Randi\'c index $R(D)$ and sum-connectivity index $\chi(D)$. First, by performing a scaling analysis we show that the average degree $\langle k \rangle$ serves as scaling parameter for the average values of $V_x(D)$, $R(D)$ and $\chi(D)$. Then, we also state expressions relating the number of arcs, spectral radius, and closed walks of length 2 to $(n,p)$, the parameters of ER random digraphs. Concerning spectral properties, we compute six different graph energies on $D(n,p)$. We start by validating $\langle k \rangle$ as the scaling parameter of the graph energies. Additionally, we reformulate a set of bounds previously reported in the literature for these energies as a function $(n,p)$. Finally, we phenomenologically state relations between energies that allow us to extend previously known bounds.

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 $p$. We give a rationale for the relevance of such "throttle/brake" coupled systems (physically open systems) to grasp the stability issues of complex networks in areas such as biochemistry, neurons or economy, for which their modelling in terms of non-hermitian Hamiltonians is still in infancy. Specifically, we measure the connectivity between the two sets with the parameter $\alpha$, 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 $H(\gamma,\alpha,N)$ of this model presents pseudo-Hermiticity, where $\gamma$ is the loss/gain strength. However, we show that for a given graph setup $H(\gamma,\alpha,N)$ becomes ${\cal PT}-$symmetric. In both scenarios (pseudo-Hermiticity and ${\cal PT}-$symmetric), depending on the parameter combination, the spectra of $H(\gamma,\alpha,N)$ can be real even when it is non-hermitian. Thus, we numerically characterize the average fractions of real and imaginary eigenvalues of $H(\gamma,\alpha,N)$ as a function of the parameter set $\{\gamma,\alpha,N\}$. We demonstrate, for both setups, that there is a well defined sector of the $\gamma\alpha-$plane (which grows with $N$) where the spectrum of $H(\gamma,\alpha,N)$ is predominantly real., Comment: 10 pages, 9 figures

We consider random geometric graphs on the plane characterized by a non-uniform density of vertices. In particular, we introduce a graph model where $n$ vertices are independently distributed in the unit disc with positions, in polar coordinates $(l,\theta)$, obeying the probability density functions $\rho(l)$ and $\rho(\theta)$. Here we choose $\rho(l)$ as a normal distribution with zero mean and variance $\sigma\in(0,\infty)$ and $\rho(\theta)$ as an uniform distribution in the interval $\theta\in [0,2\pi)$. Then, two vertices are connected by an edge if their Euclidian distance is less or equal than 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 $\left\langle k \right\rangle$ and the number of non-isolated 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 $\left\langle k(n,\sigma,\ell) \right\rangle$. Then, we look for the scaling properties of the normalized average measure $\left\langle \overline{X} \right\rangle$ (where $X$ stands for $V_\times$, $r$ and $S$) over graph ensembles. We demonstrate that the scaling parameter of $\left\langle \overline{V_\times} \right\rangle=\left\langle V_\times \right\rangle/n$ is indeed $\left\langle k \right\rangle$; with $\left\langle \overline{V_\times} \right\rangle \approx 1-\exp(-\left\langle k \right\rangle)$. Meanwhile, the scaling parameter of both $\left\langle \overline{r} \right\rangle$ and $\left\langle \overline{S} \right\rangle$ is proportional to $n^{-\gamma} \left\langle k \right\rangle$ with $\gamma\approx 0.16$., Comment: 15 pages, 14 figures

Mutualistic networks are used to study the structure and processes inherent to mutualistic relationships. In this paper, we introduce a random matrix ensemble (RME) representing the adjacency matrices of mutualistic networks composed by two vertex sets of sizes $n$ and $m-n$. Our RME depends on three parameters: the network size $n$, the size of the smaller set $m$, and the connectivity between the two sets $\alpha$, where $\alpha$ is the ratio of current adjacent pairs over the total number of possible adjacent pairs between the sets. We focus on the the spectral, eigenvector and topological properties of the RME by computing, respectively, the ratio of consecutive eigenvalue spacings $r$, the Shannon entropy of the eigenvectors $S$, and the Randi\'c index $R$. First, within a random matrix theory approach (i.e., a statistical approach), we identify a parameter $\xi\equiv\xi(n,m,\alpha)$ that scales the average normalized measures $\left< \overline{X} \right>$ (with $X$ representing $r$, $S$ and $R$). Specifically, we show that (i) $\xi\propto \alpha n$ with a weak dependence on $m$, and (ii) for $\xi<1/10$ most vertices in the mutualistic network are isolated, while for $\xi>10$ the network acquires the properties of a complete network, i.e., the transition from isolated vertices to a complete-like behavior occurs in the interval $1/10<\xi<10$. Then, we demonstrate that our statistical approach predicts reasonably well the properties of real-world mutualistic networks; that is, the universal curves $\left< \overline{X} \right>$ vs. $\xi$ show good correspondence with the properties of real-world networks.

In this work we perform computational and analytical studies of the Randi\'c index $R(G)$ in Erd\"os-R\'{e}nyi models $G(n,p)$ characterized by $n$ vertices connected independently with probability $p \in (0,1)$. First, from a detailed scaling analysis, we show that $\left\langle \overline{R}(G) \right\rangle = \left\langle R(G)\right\rangle/(n/2)$ scales with the product $\xi\approx np$, so we can define three regimes: a regime of mostly isolated vertices when $\xi < 0.01$ ($R(G)\approx 0$), a transition regime for $0.01 < \xi < 10$ (where $0 10$ ($R(G)\approx n/2$). Then, motivated by the scaling of $\left\langle \overline{R}(G) \right\rangle$, we analytically (i) obtain new relations connecting $R(G)$ with other topological indices and characterize graphs which are extremal with respect to the relations obtained and (ii) apply these results in order to obtain inequalities on $R(G)$ for graphs in Erd\"os-R\'{e}nyi models., Comment: 28 pages, 10 figures

Bipartite graphs are often found to represent the connectivity between the components of many systems such as ecosystems. A bipartite graph is a set of $n$ nodes that is decomposed into two disjoint subsets, having $m$ and $n-m$ vertices each, such that there are no adjacent vertices within the same set. The connectivity between both sets, which is the relevant quantity in terms of connections, can be quantified by a parameter $\alpha\in[0,1]$ that equals the ratio of existent adjacent pairs over the total number of possible adjacent pairs. Here, we study the spectral and localization properties of such random bipartite graphs. Specifically, within a Random Matrix Theory (RMT) approach, we identify a scaling parameter $\xi\equiv\xi(n,m,\alpha)$ that fixes the localization properties of the eigenvectors of the adjacency matrices of random bipartite graphs. We also show that, when $\xi<1/10$ ($\xi>10$) the eigenvectors are localized (extended), whereas the localization--to--delocalization transition occurs in the interval $1/10<\xi<10$. Finally, given the potential applications of our findings, we round off the study by demonstrating that for fixed $\xi$, the spectral properties of our graph model are also universal., Comment: 17 pages, 10 figures