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It is the maximum of the degree of a vertex over all vertices.

The minimum degree is $\delta (G)={\min }_{v\in V(G)}\mathrm{deg}(v)$, the least degree of a vertex in the graph.

A universal vertex is adjacent to every other vertex of its graph.

For a finite undirected graph with $m>0$ edges, the degree-biased probability distribution isChoose an edge uniformly, then choose one of its two endpoint incidences uniformly. A vertex is selected once for each incident edge end, so the handshaking lemma normalizes the probabilities. For the three-vertex path graph, the probabilities are $(1/4,1/2,1/4)$.

In a bipartite graph, choosing only the endpoint in one specified part instead gives $\mathrm{deg}(v)/m$ on that part and zero on the other. This conditional sampling convention differs from the probability distribution on the whole vertex set.

Projective degree belongs to an embedding rather than to the abstract variety. A line and a smooth conic in ${\mathbb{P}}^2$ are both abstractly ${\mathbb{P}}^1$ but have degrees one and two.

Adjoining projective coordinates that vanish identically does not change the homogeneous coordinate ring or Hilbert polynomial. A linear embedding of projective space therefore preserves the degree of an embedded variety.

For a general linear form $\ell$ not vanishing identically on a projective curve, the exact sequenceshows that the hyperplane-section length is $P_C(m)-P_C(m-1)=\mathrm{deg}C$.

The Hilbert function of a finitely generated graded algebra agrees in all sufficiently large degrees with a unique polynomial. For a projective variety of dimension $r$, its leading term is $\mathrm{deg}(V)m^r/r!$.

A nonzero polynomial of degree $d$ over a field has at most $d$ distinct roots. Indeed, each root supplies a linear factor by polynomial division, and induction on $d$ gives the bound.

If $S$ is compact and $V$ is a smooth vector field, thenis smooth and well-defined by the Hopf-Rinow theorem. The maps ${\varphi }_t(p)={\exp }_p(tV(p))$ form a smooth homotopy from the identity to $\varphi$, so ${\mathrm{deg}}_2(\varphi )=1$.

The constructible hierarchy is defined byfor limit ordinals $\lambda$. Its union is the constructible universe $L$.

For a quadratic field $K$ of discriminant $D$,where ${\chi }_D(p)$ is $1$, $-1$, or $0$ according as $(p)$ splits, remains prime, or ramifies in ${\mathcal{O}}_K$. Comparing the three possible local Euler factors proves the formal identity.

Unique factorization of nonzero ideals and multiplicativity of the ideal norm give the formal identitywhere the product ranges over nonzero prime ideals.

The maximal ideal of an integrally closed Noetherian local domain of Krull dimension one is principal. Hence every such local ring is a discrete valuation ring.

In a prefix code no codeword is the prefix of another, so concatenated messages have unique instantaneous decoding.

A collisional cascade transfers mass from large bodies to progressively smaller fragments. A steady cascade has a size distribution whose mass flux through logarithmic size bins is independent of size.

The fractional luminosity is $f=L_{\mathrm{disk}}/L_{\ast }$; for blackbody grains in an optically thin belt at radius $r$, it equals their total geometric cross-section divided by $4\pi r^2$.

The halo model represents matter as a population of dark matter halos with specified profiles, abundance and clustering. Pairs of mass elements inside the same halo generate the one-halo term; pairs in distinct halos generate the two-halo term. This connects internal object structure with large-scale statistics. The decomposition is useful, but assumptions about halo boundaries and correlations make the simplest implementation approximate.^[ref]

The peak-background split decomposes density fluctuations into long- and short-wavelength parts. A long overdensity lowers the effective local collapse threshold and thereby changes the abundance of halos.

The spherical-collapse model applies a spherical top-hat filter to an overdense region and predicts collapse when its linearly extrapolated smoothed overdensity exceeds the threshold ${\delta }_c\simeq 1.686$ in an Einstein-de Sitter universe.