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The adjacency spectral radius lies between the average degree and the maximum degree

Statement

Let $G$ be a finite simple graph on $n \geq 1$ vertices, let

$\overset{ˉ}{d} (G) : = \frac{1}{n} \sum_{v \in V (G)} \deg (v)$

be its average degree, and let $Δ (G)$ be its maximum degree. Then its adjacency spectral radius satisfies

$\overset{ˉ}{d} (G) \leq ρ (G) \leq Δ (G) .$

Facts & Assumptions

Given: A finite simple graph $G$ on $n \geq 1$ vertices, with adjacency matrix $A$ .

[F1]

The degree of a vertex is the number of $1$ 's in the corresponding row of $A$ , and $Δ (G)$ is the maximum of those row sums (Adjacency, incidence, open and closed neighbourhoods, vertex degree, minimum degree and maximum degree).

[L1]

For a real symmetric matrix, the largest eigenvalue is the maximum of the Rayleigh quotient (The smallest and largest eigenvalues of a self-adjoint endomorphism are the minimum and maximum Rayleigh quotients, The Rayleigh quotient of a nonzero vector for a self-adjoint endomorphism).

[L2]

Every eigenvalue of a matrix lies in one of its Gershgorin disks (Every eigenvalue lies in some Gershgorin disk).

[F2]

The spectral radius is the maximum absolute value of the adjacency eigenvalues (Adjacency spectrum, spectral radius, and cospectral graphs).

Proof

technique · direct

1.1

Let $1$ be the all-ones vector. Then $R_{A} (1) = \frac{1^{T} A 1}{1^{T} 1} = \frac{\sum_{v \in V (G)} \deg (v)}{n} = \overset{ˉ}{d} (G)$ , because $1^{T} A 1$ adds all row sums of $A$ . By [L1], the largest adjacency eigenvalue is at least this Rayleigh quotient, so $λ_{1} (G) \geq \overset{ˉ}{d} (G)$ . Hence $ρ (G) \geq \overset{ˉ}{d} (G)$ by [F2].

F1L1F2algebra

1.2

Every Gershgorin disk of $A$ is centered at $0$ , because the diagonal of an adjacency matrix is zero, and its radius is the corresponding row sum, hence at most $Δ (G)$ by [F1]. Therefore [L2] implies $∣ λ ∣ \leq Δ (G)$ for every adjacency eigenvalue $λ$ , so [F2] gives $ρ (G) \leq Δ (G)$ .

F1L2F2

2.1

Combining steps 1.1 and 1.2 yields the stated bounds.

step 1.1step 1.2∎

Depends on

Used by

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Sources

O. Pikhurko, Algebraic Methods in Combinatorics, Section 14.1 (standard reference, not scraped)