arXiv subjects

Jose M. Sigarreta

Publications and source records attributed to Jose M. Sigarreta.

12 recordsLinked to original sources

arXiv2023

Multiplicative topological indices: Analytical properties and application to random networks

We make use of multiplicative degree-based topological indices $X_Π(G)$ to perform a detailed analytical and statistical study of random networks $G=(V(G),E(G))$. We consider two classes of indices: $X_Π(G) = \prod_{u \in V(G)} F_V(d_u)$ and $X_Π(G) = \prod_{uv \in E(G)} F_E(d_u,d_v)$, where $uv$ denotes the edge of $G$ connecting the vertices $u$ and $v$, $d_u$ is the degree of the vertex $u$, and $F_V(x)$ and $F_E(x,y)$ are functions of the vertex degrees. Specifically, we find analytical inequalities involving these multiplicative indices. Also, we apply $X_Π(G)$ on three models of random networks: Erdös-Rényi networks, random geometric graphs, and bipartite random networks. We show that $\left< \ln X_Π(G) \right>$, normalized to the order of the network, scale with the corresponding average degree; here $\left< \cdot \right>$ denotes the average over an ensemble of random networks.

math.CO↗

arXiv2022

Revan-degree indices on random graphs

Given a simple connected non-directed graph $G=(V(G),E(G))$, we consider two families of graph invariants: $RX_Σ(G) = \sum_{uv \in E(G)} F(r_u,r_v)$ (which has gained interest recently) and $RX_Π(G) = \prod_{uv \in E(G)} F(r_u,r_v)$ (that we introduce in this work); where $uv$ denotes the edge of $G$ connecting the vertices $u$ and $v$, $r_u$ is the Revan degree of the vertex $u$, and $F$ is a function of the Revan vertex degrees. Here, $r_u = Δ+ δ- d_u$ with $Δ$ and $δ$ the maximum and minimum degrees among the vertices of $G$ and $d_u$ is the degree of the vertex $u$. Particularly, we apply both $RX_Σ(G)$ and R$X_Π(G)$ on two models of random graphs: Erdös-Rényi graphs and random geometric graphs. By a thorough computational study we show that $\left< RX_Σ(G) \right>$ and $\left< \ln RX_Π(G) \right>$, normalized to the order of the graph, scale with the average Revan degree $\left< r \right>$; here $\left< \cdot \right>$ denotes the average over an ensemble of random graphs. Moreover, we provide analytical expressions for several graph invariants of both families in the dense graph limit.

math.CO↗

arXiv2021

Stolarsky-Puebla index

We introduce a degree-based variable topological index inspired on the Stolarsky mean (known as the generalization of the logarithmic mean). We name this new index as the Stolarsky-Puebla index: $SP_α(G) = \sum_{uv \in E(G)} d_u$, if $d_u=d_v$, and $SP_α(G) = \sum_{uv \in E(G)} \left[\left( d_u^α-d_v^α\right)/\left( α(d_u-d_v\right)\right]^{1/(α-1)}$, otherwise. Here, $uv$ denotes the edge of the network $G$ connecting the vertices $u$ and $v$, $d_u$ is the degree of the vertex $u$, and $α\in \mathbb{R} \backslash \{0,1\}$. Indeed, for given values of $α$, the Stolarsky-Puebla index reproduces well-known topological indices such as the reciprocal Randic index, the first Zagreb index, and several mean Sombor indices. Moreover, we apply these indices to random networks and demonstrate that $\left< SP_α(G) \right>$, normalized to the order of the network, scale with the corresponding average degree $\left< d \right>$.

math.CO↗

arXiv2021

Normalized Sombor indices as complexity measures of random graphs

We perform a detailed computational study of the recently introduced Sombor indices on random graphs. Specifically, we apply Sombor indices on three models of random graphs: Erdös-Rényi graphs, random geometric graphs, and bipartite random graphs. Within a statistical random matrix theory approach, we show that the average values of Sombor indices, normalized to the order of the graph, scale with the graph average degree. Moreover, we discuss the application of average Sombor indices as complexity measures of random graphs and, as a consequence, we show that selected normalized Sombor indices are highly correlated with the Shannon entropy of the eigenvectors of the graph adjacency matrix.

math.CO↗

arXiv2021

Analytical and computational study of the variable inverse sum deg index

A large number of graph invariants of the form $\sum_{uv \in E(G)} F(d_u,d_v)$ are studied in mathematical chemistry, where $uv$ denotes the edge of the graph $G$ connecting the vertices $u$ and $v$, and $d_u$ is the degree of the vertex $u$. Among them the variable inverse sum deg index $ISD_a$, with $F(d_u,d_v)=1/(d_u^a+d_v^a)$, was found to have applicative properties. The aim of this paper is to obtain new inequalities for the variable inverse sum deg index, and to characterize graphs extremal with respect to them. Some of these inequalities generalize and improve previous results for the inverse sum deg index. In addition, we computationally validate some of the obtained inequalities on ensembles of random graphs and show that the ratio $\left\langle ISD_a(G) \right\rangle/n$ ($n$ being the order of the graph) depends only on the average degree $\left\langle d \right\rangle$.

math.CO↗

arXiv2021

Analytical and computational properties of the variable symmetric division deg index

The aim of this work is to obtain new inequalities for the variable symmetric division deg index $SDD_α(G) = \sum_{uv \in E(G)} (d_u^α/d_v^α+d_v^α/d_u^α)$, and to characterize graphs extremal with respect to them. Here, $uv$ denotes the edge of the graph $G$ connecting the vertices $u$ and $v$, $d_u$ is the degree of the vertex $u$, and $α\in \mathbb{R}$. Some of these inequalities generalize and improve previous results for the symmetric division deg index. In addition, we computationally apply the $SDD_α(G)$ index on random graphs and show that the ratio $\left\langle SDD_α(G) \right\rangle/n$ ($n$ being the order of the graph) depends only on the average degree $\left\langle d \right\rangle$.

math.CO↗

arXiv2020

Topological versus spectral properties of random geometric graphs

In this work we perform a detailed statistical analysis of topological and spectral properties of random geometric graphs (RGGs); a graph model used to study the structure and dynamics of complex systems embedded in a two dimensional space. RGGs, $G(n,\ell)$, consist of $n$ vertices uniformly and independently distributed on the unit square, where two vertices are connected by an edge if their Euclidian distance is less or equal than the connection radius $\ell \in [0,\sqrt{2}]$. To evaluate the topological properties of RGGs we chose two well-known topological indices, the Randić index $R(G)$ and the harmonic index $H(G)$. While we characterize the spectral and eigenvector properties of the corresponding randomly-weighted adjacency matrices by the use of random matrix theory measures: the ratio between consecutive eigenvalue spacings, the inverse participation ratios and the information or Shannon entropies $S(G)$. First, we review the scaling properties of the averaged measures, topological and spectral, on RGGs. Then we show that: (i) the averaged--scaled indices, $\left\langle R(G) \right\rangle$ and $\left\langle H(G) \right\rangle$, are highly correlated with the average number of non-isolated vertices $\left\langle V_\times(G) \right\rangle$; and (ii) surprisingly, the averaged--scaled Shannon entropy $\left\langle S(G) \right\rangle$ is also highly correlated with $\left\langle V_\times(G) \right\rangle$. Therefore, we suggest that very reliable predictions of eigenvector properties of RGGs could be made by computing topological indices.

cond-mat.dis-nn↗

arXiv2020

Computational and analytical studies of the Randić index in Erdös-Rényi models

In this work we perform computational and analytical studies of the Randić index $R(G)$ in Erdös-Rényi 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 $ξ\approx np$, so we can define three regimes: a regime of mostly isolated vertices when $ξ< 0.01$ ($R(G)\approx 0$), a transition regime for $0.01 < ξ< 10$ (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 Erdös-Rényi models.

cond-mat.dis-nn↗

arXiv2019

Spectral and localization properties of random bipartite graphs

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 $α\in[0,1]$ 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 $ξ\equivξ(n,m,α)$ that fixes the localization properties of the eigenvectors of the adjacency matrices of random bipartite graphs. We also show that, when $ξ<1/10$ ($ξ>10$) the eigenvectors are localized (extended), whereas the localization--to--delocalization transition occurs in the interval $1/10<ξ<10$. Finally, given the potential applications of our findings, we round off the study by demonstrating that for fixed $ξ$, the spectral properties of our graph model are also universal.

cond-mat.dis-nn↗

arXiv2013

The metric dimension of strong product graphs

For an ordered subset $S = \{s_1, s_2,\dots s_k\}$ of vertices and a vertex $u$ in a connected graph $G$, the metric representation of $u$ with respect to $S$ is the ordered $k$-tuple $ r(u|S)=(d_G(v,s_1), d_G(v,s_2),\dots,$ $d_G(v,s_k))$, where $d_G(x,y)$ represents the distance between the vertices $x$ and $y$. The set $S$ is a metric generator for $G$ if every two different vertices of $G$ have distinct metric representations. A minimum metric generator is called a metric basis for $G$ and its cardinality, $dim(G)$, the metric dimension of $G$. It is well known that the problem of finding the metric dimension of a graph is NP-Hard. In this paper we obtain closed formulae and tight bounds for the metric dimension of strong product graphs.

math.CO↗

arXiv2010

Partitioning a graph into defensive k-alliances

A defensive $k$-alliance in a graph is a set $S$ of vertices with the property that every vertex in $S$ has at least $k$ more neighbors in $S$ than it has outside of $S$. A defensive $k$-alliance $S$ is called global if it forms a dominating set. In this paper we study the problem of partitioning the vertex set of a graph into (global) defensive $k$-alliances. The (global) defensive $k$-alliance partition number of a graph $Γ=(V,E)$, ($ψ_{k}^{gd}(Γ)$) $ψ_k^{d}(Γ)$, is defined to be the maximum number of sets in a partition of $V$ such that each set is a (global) defensive $k$-alliance. We obtain tight bounds on $ψ_k^{d}(Γ)$ and $ψ_{k}^{gd}(Γ)$ in terms of several parameters of the graph including the order, size, maximum and minimum degree, the algebraic connectivity and the isoperimetric number. Moreover, we study the close relationships that exist among partitions of $Γ_1\times Γ_2$ into (global) defensive $(k_1+k_2)$-alliances and partitions of $Γ_i$ into (global) defensive $k_i$-alliances, $i\in \{1,2\}$.

math.CO↗

arXiv2009

On global offensive k-alliances in graphs

We investigate the relationship between global offensive $k$-alliances and some characteristic sets of a graph including $r$-dependent sets and $τ$-dominating sets. As a consequence of the study, we obtain bounds on the global offensive $k$-alliance number in terms of several parameters of the graph.

math.CO↗