Enlarging a connected graph while keeping entropy and spectral radius: self-similarity techniques

TL;DR

This work formalizes graph self-similarity through orbital similarity, introducing seeds, orbit partitions, and the orbit divisor matrix $S_G$. It proves that orbitally similar graphs share the same entropy ${\rm Ent}(G)$ and spectral radius ${\varrho}(G)$, and develops constructive methods to generate infinite self-similar sequences via prism, strong-prism, corona, and loading operations, preserving orbital structure. The authors demonstrate that self-similarity preserves a broad set of invariants (including ${\rm Ent}$ and ${\varrho}$) while outlining how density, cyclomatic number, and other properties evolve, depending on the seed (trees vs. unicyclic vs. multicyclic). They provide concrete families (cycles, prisms, torus grids, sun graphs, and loaded torii) as examples and establish general principles for building new self-similar sequences from existing ones, with a detailed treatment of vertex-transitive and bi-orbital seeds. The results have potential applications in network design and spectral graph analysis where scalable, invariant-rich graph growth is desirable.

Abstract

This work is about self-similar sequences of growing connected graphs. We explain how to construct such sequences and why they are important. We show for instance that all the connected graphs in a self-similar sequence have not only the same entropy, but also the same spectral radius.

Figures (11)

• Figure 1: Orbitally similar graphs. In particular, they have equal entropy and equal spectral radius.
• Figure 2: Vertex-transitivity, asymmetry, and partial symmetry.
• Figure 3: A sample of connected graphs with $3$ orbits each.
• Figure 4: Both graphs have the same orbit divisor matrix.
• Figure 5: Orbitally similar non-isomorphic graphs of the same order.

Theorems & Definitions (43)

• Proposition 1
• Example 1
• Example 2
• Definition 1
• Proposition 2
• proof
• Definition 2
• Theorem 1
• proof
• Example 3