[Paper Review] Operations on Graphs Increasing Some Graph Parameters

Alexander Kelmans|arXiv (Cornell University)|Sep 21, 2011

Advanced Graph Theory Research15 references3 citations

TL;DR

This paper investigates graph operations that increase key graph parameters such as Laplacian eigenvalues, reliability, and number of spanning trees. It introduces symmetric operations on weighted graphs that preserve or enhance these parameters under specific conditions, proving that certain transformations increase the largest Laplacian eigenvalue in bipartite graphs, thereby establishing monotonicity in reliability and spectral properties across graph families.

ABSTRACT

In this partly expository paper we discuss and describe some of our old and recent results on partial orders on the set (m,n)-graphs (i.e. graphs with n vertices and m edges) and some operations on graphs that are monotone with respect to these partial orders. The partial orders under consideration include those related with some Laplacian characteristics of graphs as well as with some probabilistic characteristics of graphs with randomly deleted edges. Section 2 provides some notions, notation, and simple observations. Section 3 contains some basic facts on the Laplacian polynomial of a graph. Section 4 describes various graph operation and their properties. In Section 5 we introduce some partial orders on the set of (m,n)-graphs related, in particular, with the graph Laplacian and the graph reliability (Laplacian posets and reliability posets}). Section 6 contains some old and recent results on the monotonicity of some graph operations with respect to Laplacian posets. Section 7 and 8 include some old and recent results on the monotonicity of some graph operations with respect to reliability posets and to some other parameters of graphs as well as some open problems. Section 9 contains some generalizations of the described results on weighted graphs. Keywords: graph, graph operations, graph posets, random graphs, decomposable graphs, threshold graphs, weighted graphs, spanning subgraphs, Laplacian polynomial and spectrum, adjacency polynomial and spectrum, graph reliability, trees, forests, Hamiltonian cycle and path, symmetric polynomials

Motivation & Objective

• To study operations on graphs that increase fundamental graph parameters such as the number of spanning trees, reliability, and Laplacian eigenvalues.
• To define and analyze partial orders on graphs with fixed vertex and edge counts based on reliability and Laplacian spectral properties.
• To generalize results on graph operations from unweighted to weighted graphs, particularly focusing on symmetric (v₁,v₂,ε)-operations.
• To investigate whether most reliable or least reliable graphs exist independently of edge failure probability p, and whether such graphs are unique.
• To extend threshold graph concepts to weighted graphs and establish spectral monotonicity under symmetric edge-weight operations.

Proposed method

• Introduces a symmetric (v₁,v₂,ε)-operation on weighted graphs that redistributes edge weights between two vertices while preserving total edge weight.
• Uses the Perron-Frobenius theorem and Courant-Swartz min-max principle to analyze the largest eigenvalue β(Kʷ) of the adjacency matrix of the weighted graph.
• Applies the Matrix Tree Theorem and Laplacian matrix properties to relate the number of spanning trees to the eigenvalues of the Laplacian matrix.
• Defines reliability R(p,G) as the probability that a random subgraph with edge retention probability p remains connected, and compares graphs via the partial order G ⪰ʳ F when R(p,G) ≥ R(p,F) for all p ∈ [0,1].
• Introduces the concept of bipartite weighted graphs via the condition bip(Kʷ) = odc(Kʷ) = n+1, and proves that symmetric operations preserve or increase the largest Laplacian eigenvalue λ(Kʷ) in such graphs.
• Generalizes threshold graphs to weighted graphs by defining x ⪯ y if w(xv) ≤ w(yv) for all v ≠ x,y, and proves monotonicity of λ(Kʷ) under symmetric operations on such graphs.

Experimental results

Research questions

• RQ1 Does every set of graphs with n vertices and m edges contain a most reliable graph that is independent of the edge failure probability p?
• RQ2 Can symmetric (v₁,v₂,ε)-operations on weighted graphs be shown to increase the largest Laplacian eigenvalue when the resulting graph is bipartite?
• RQ3 Under what conditions does a symmetric (v₁,v₂,ε)-operation preserve or increase the number of spanning trees in a graph?
• RQ4 Is the reliability order ⪰ʳ a total order, or are there incomparable graphs in Gₙᵐ?
• RQ5 Can the concept of threshold graphs be extended to weighted graphs such that symmetric operations preserve or increase spectral parameters?

Key findings

• A symmetric (v₁,v₂,ε)-operation on a weighted graph Kʷ increases the largest eigenvalue β(Kʷ) of the adjacency matrix if the operation is symmetric and the weight function satisfies w(v₁s) ≥ w(v₂s) ≥ ε(s) ≥ 0 for all s in the common neighborhood.
• For a bipartite weighted graph Kʷ, a symmetric (v₁,v₂,ε)-operation increases the largest Laplacian eigenvalue λ(Kʷ), provided the operation preserves bipartiteness.
• If G′ is obtained from G by a symmetric Kₓᵧ-operation and G′ is bipartite, then G ⪰ᶠ G′, meaning the reliability order is preserved under such operations.
• The condition bip(Kʷ) = odc(Kʷ) = n+1 characterizes bipartite weighted graphs, and under this condition, L(λ,Kʷ) = L⁺(λ,Kʷ), ensuring equality of Laplacian and signless Laplacian characteristic polynomials.
• The largest eigenvalue λ(Kʷ) of the Laplacian matrix increases under symmetric (v₁,v₂,ε)-operations on weighted threshold graphs, where x ⪯ y holds for all pairs of vertices.
• The paper proves that for certain (n,m) pairs, no ⪰ʳ-maximum graph exists in Gₙᵐ, implying that the most reliable graph depends on p, and thus the reliability order is not total.