On the spectral radius of a directed graph

Kwapisz, Jarosław

doi:10.1002/(sici)1097-0118(199612)23:4<405::aid-jgt9>3.0.co;2-v

Cited by 9 publications

(2 citation statements)

References 2 publications

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“…The maximum spectral radius for an undirected graph with a given N , m is as follows:

which reduces to N − 1 for a fully connected (complete) graph. For a digraph:

or:

where

indicate the maximum in and out degrees (number of incoming or outgoing links or edges) from any node [ 28 ].…”

Section: Methodsmentioning

confidence: 99%

Embedded Complexity of Evolutionary Sequences

Phillips

2024

Entropy

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Multiple pathways and outcomes are common in evolutionary sequences for biological and other environmental systems due to nonlinear complexity, historical contingency, and disturbances. From any starting point, multiple evolutionary pathways are possible. From an endpoint or observed state, multiple possibilities exist for the sequence of events that created it. However, for any observed historical sequence—e.g., ecological or soil chronosequences, stratigraphic records, or lineages—only one historical sequence actually occurred. Here, a measure of the embedded complexity of historical sequences based on algebraic graph theory is introduced. Sequences are represented as system states S(t), such that S(t − 1) ≠ S(t) ≠ S(t + 1). Each sequence of N states contains nested subgraph sequences of length 2, 3, …, N − 1. The embedded complexity index (which can also be interpreted in terms of embedded information) compares the complexity (based on the spectral radius λ1) of the entire sequence to the cumulative complexity of the constituent subsequences. The spectral radius is closely linked to graph entropy, so the index also reflects information in the sequence. The analysis is also applied to ecological state-and-transition models (STM), which represent observed transitions, along with information on their causes or triggers. As historical sequences are lengthened (by the passage of time and additional transitions or by improved resolutions or new observations of historical changes), the overall complexity asymptotically approaches λ1 = 2, while the embedded complexity increases as N2.6. Four case studies are presented, representing coastal benthic community shifts determined from biostratigraphy, ecological succession on glacial forelands, vegetation community changes in longleaf pine woodlands, and habitat changes in a delta.

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“…The maximum spectral radius for an undirected graph with a given N , m is as follows:

which reduces to N − 1 for a fully connected (complete) graph. For a digraph:

or:

where

indicate the maximum in and out degrees (number of incoming or outgoing links or edges) from any node [ 28 ].…”

Section: Methodsmentioning

confidence: 99%

Embedded Complexity of Evolutionary Sequences

Phillips

2024

Entropy

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“…In 2022, Shan et al [115] characterized the extremal digraphs with the maximal or minimal α-spectral radius among some digraph classes, such as rose digraphs, generalized theta digraphs, and tri-ring digraphs with given size m. In 2023, Ganie and Carmona [41] established an increasing sequence of lower bounds for the spectral radius of digraphs. For more results on digraphs see [4,10,39,40,42,55,59,64,65,71,82,89,128,[130][131][132]. In this thesis, we mainly study the spectral radii of the A α -matrix and the eccentricity matrix of digraphs.…”

Section: Spectral Radiusmentioning

confidence: 99%

Spectral properties of digraphs with a fixed dichromatic number

Yang

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Other papers by the author[1] Ordering of bicyclic signed digraphs by energy, Filomat 34 (2020), 4297-4309 (with L. Wang).[2] Iota energy orderings of bicyclic signed digraphs, Transactions on Combinatorics 10 (2021), 187-200 (with L. Wang).[3] The eccentricity matrix of a digraph, Discrete Applied Mathematics 322 (2022), 61-73 (with L. Wang).[4] On the sum of the k largest absolute values of Laplacian eigenvalues of digraphs,

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Complexity of Earth Surface System Evolutionary Pathways

Phillips

2016

Math Geosci

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Evolution of Earth surface systems (ESS) comprises sequential transitions between system states. Treating these as directed graphs, algebraic graph theory was used to quantify complexity of archetypal structures, and empirical examples of forest succession and alluvial river channel change. Spectral radius measures structural complexity and is highest for fully connected, lowest for linear sequential and cyclic graphs, and intermediate for divergent and convergent patterns. The irregularity index β represents the extent to which a subgraph is representative of the full graph. Fully connected graphs have β = 1. Lower values are found in linear and cycle patterns, while higher values, such as those of divergent and convergent patterns, are due to a few highly connected nodes. Algebraic connectivity (μ(G)) indicates inferential synchronization and is inversely related to historical contingency. Highest values are associated with fully connected and strongly connected mesh graphs, whereas forking structures and linear sequences all have μ(G) = 1, with cycles slightly higher. Diverging vs. converging graphs of the same size and topology have no differences with respect to graph complexity, so complexity change is dependent on whether development results in increased or reduced richness. Convergent-divergent mode switching, however, would generally increase ESS complexity, decrease irregularity, and increase algebraic connectivity. As ESS and associated graphs evolve, none of the possible trends reduces complexity, which can only remain constant or increase. Algebraic connectivity may increase, however. As improving shortcomings in ESS 123 Math Geosci evolution models generally result in elaborating possible state changes, this produces more structurally complex but less historically contingent models.

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On the spectral radius of a directed graph

Abstract: We provide upper estimates on the spectral radius of a directed graph. In particular we prove that the spectral radius is bounded by the maximum of the geometric mean of in‐degree and out‐degree taken over all vertices. © 1996 John Wiley & Sons, Inc.

Cited by 9 publications

References 2 publications

Embedded Complexity of Evolutionary Sequences

Embedded Complexity of Evolutionary Sequences

Spectral properties of digraphs with a fixed dichromatic number

Complexity of Earth Surface System Evolutionary Pathways

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