Intrinsic Metrics on Graphs & Graph Geometry D. J. Klein Texas - PDF document

Intrinsic Metrics on Graphs & Graph Geometry D. J. Klein Texas A&M University @ Galveston, Galveston, TX 77553-1675 kleind@tamug.edu Abstract Graphs are a cosmopolitan representation of a wide range of things: group networks in sociology, food-webs in eco-biology, reaction networks in biochemistry, Feynman diagrams in physics, electrical circuits in engineering & physics, and molecules in chemistry, & molecular biology. As molecular representations, graphs seemingly retain information about only a small part of a molecule ’ s character – in particular, they seem to suppress molecular geometry (along with associated electron densities), though ever since their introduction ~150 years ago, they have been extensively utilized. Thus the intrinsic characteristics of graphs are of general interest. Naturally there is a question of intrinsic metrics on graphs, independently of whether the graphs are used to represent molecules, or whatever. In mathematical graph-theory, there is extensive work on the shortest- path metric – and practically no other metric. Still one might imagine other possibilities for an intrinsic graph metric, such as we address here, along with some ideas for uses of such further metrics.

The ideas & work reported here, owe much to several colleagues at TAMU@G as well as numerous visitors from around the world – including: WA Seitz, TG Schmalz, M Randic, N Trinajstic, NH March, Y Yang, AT Balaban, & very many more Also very important has been grant support especially from the Welch Foundation of Houston, Texas

G ≡ graph * G is a set V of vertices & set E of edges between unordered pairs of vertices ⎧ molecules in chemistry ⎪ electric circuits in physics & electrical engineering ⎪ ( ) ⎪ Feynman interaction diagrams in physics ⎪ ⎪ reaction networks in chemistry & biochemistry ⎪ respiratory & circulatory networks in biology ⎪ ⎪ food webs in eco-biology ⎪ ⎨ * realized as: migratory routes in ecology ⎪ psychological & sociological interaction patterns ⎪ ⎪ transportation networks in geography & business ⎪ evolutionary patterns in biology ⎪ ⎪ neural networks (as in the brain) ⎪ ( ) ⎪ communication networks like www ⎪ ⎩ etc * graphs are ubiquitous with numerous mathematical papers, & texts

Graph Distances * Is there an intrinsic geometry of graphs? * Is there an intrinsic distance function on graphs? * If there is more than one, what might be their characteristics & differences? * Are there natural graph invariants with a geometric interpretation? ρ × → × ρ � � * distance functions (or metrics) on graphs ( : V V : i j ( , ) i j ) ρ = example ( D ): ≡ ( , ) { } D i j chemical distance ≡ { - } through bond distance ≡ { } topological distance ≡ { - } shortest path distance ≡ → {minimum # edges along path } i j

Lots of work − most only on shortest-path metric − as in the >300 page book: * Are there any other intrinsic graph metrics?

Ω Candidate combinatorial metric comb * define a tree as a connected acyclic graph * define a bi-tree as an acyclic graph with 2 components * a subgraph of G is spanning if it includes all vertices of G (# spanning bi-trees, & in different components) i j Ω ≡ ( , ) i j comb (# spanning trees) * the denominator (# spanning trees) is just a normalization * in the numerator, such spanning bi-trees can be generated from spanning trees by deleting an edge “between” i & j – it being more or less plausible that the farther apart i & j are, the more choices for such a deletion (and consequent bi-tree) there will be

Ω Candidate random-walk metric walk ≡ ∈ V * define d (degree of vertex i ) i ⎛ ⎞ probability of a random walk leaving i = ⎜ * 1/ d ⎟ i ⎝ going to any particular one of its neighbors ⎠ ⎛ ⎞ probability of a random walk → ≡ ⎜ * define p i ( j ) ⎟ ⎝ leaving to arrive at before ⎠ i j i Ω ≡ → walk ( , ) / ( ) i j d p i j j Ω → * the factor just makes walk ( , ) symmetric in its arguments – since ( ) d i j p i j j involves a probability 1 / at its first step, but no similar 1/ for its arriving site j d d i j * the plausibility of this as a metric occurs since the farther apart i & j are, the less likely it should be for a walker from to end up at j before returning to i i

Ω Candidate wave-amplitude metric wave − ∈ ⎧ ⎫ 1 , { , } i j E ⎛ ⎞ ⎪ ⎪ discretized version of ≡ = ⇒ = ⎨ ⎬ * L d , i j L ⎜ ⎟ ij i ⎝ Laplacian operator ⎠ ⎪ ⎪ ⎩ 0 , otherwise ⎭ ψ with energy ε : ψ = εψ * (discretized) standing waves L ε ε ε * longer wave-length waves should be lower energy ε > 0 ∑ Ω ≡ ψ − ψ ε 2 wave ( , ) | | / i j ε ε i j ε * this is a correlation function on wave amplitudes, weighting the low-energy (long-wavelength) standing wave patterns more heavily – it is plausibly a metric since the nearer 2 sites i & j are, the more similar should be the corresponding longer-wavelength ψ & ψ amplitudes ε ε i j

Ω Candidate electric metric elec * let G correspond to an electric network, with unit resistors placed on each edge ⎧ ⎫ ⎪ effective resistance between & ⎪ i j Ω ≡ ( , ) i j ⎨ ⎬ elec ⎪ ⎪ (when battery connected between & ) i j ⎩ ⎭ * plausibly a metric since the farther apart i & j are, the greater (one imagines) should be the effective resistance between i & j

Ω Candidate linear-algebraic metric lin-alg − ∈ ⎧ ⎫ 1 , { , } i j E ⎪ ⎪ ( ) ≡ = ⇒ = ⎨ ⎬ * L , L "Laplacian" matrix d i j ij i ⎪ ⎪ ⎩ 0 , otherwise ⎭ * L nonnegative definite since for arbitrary vector x ∈ E ∑ ∑ = = − † * 2 L x L x | x x | x x i ij j i j i j , { , } i j * ( G connected) ⇒ (one 0-eigenvalue with eigenvector φ , all components φ equal) i ⎛ ⎞ generalized inverse of L Γ ≡ ⎜ * ⎟ φ φ ⎝ ⎠ 0 on -space & inverse of on space orthogonal to L Ω = Γ −Γ −Γ +Γ lin-alg ( , ) i j ii ij ji jj * perhaps a metric because L correlates to “neighborliness”, so that one expects the “inverse” Γ plausibly correlates to “non-neighborliness” (or distance)

Metrics ? Ω = ⎧ ⎫ ( , ) 0 i i ⎪ ⎪ Ω × → ⇔ Ω = Ω > ≠ ∈ � ⎨ ⎬ * ( : metric) ( , ) ( , ) 0, , , V V i j j i i j i j V ⎪ ⎪ Ω + Ω ≥ Ω ⎩ ( , ) ( , ) ( , ) ⎭ i j j k i k Ω Ω Ω Ω Ω * Are any of , , , , metrics? wave comb walk elec lin-alg Yes! All of them are metrics! & All of them are the same Ω ! Ω = Ω old: ~~~ G. Kirchoff, Ann. Phys. Chem. 72 (1847) 497-501 elec lin-alg Ω = Ω P. G. Doyle & L. J. Snell, Random Walks & Electric Networks (MAA, 1984). elec walk Ω = Ω L. W. Shapiro, Math. Mag. 60 (1987) 36-38. elec comb Ω ≡ Ω = metric DJK & M. Randic, J. Math. Chem. 12 (1993) 81-95. elec G.E. Sharpe, Electron. Lett. 3 (1967) 444–445 & 543–544. A.D. Gvishiani, V.A. Gurvich,, Russ. Math. Surveys 42 (1987) 235–236. Ω = Ω DJK, Commun. Math. Chem. (MATCH) 35 (1996) 7-27. lin-alg wave

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