The extremal α-index of outerplanar and planar graphs

Abstract

Let G be a graph of order n with adjacency matrix A(G) and let D(G) be the diagonal matrix of the degrees of G. For any real α ∈ [0, 1], write A_α(G) for the matrix

$$A_{\alpha }\left(G\right)=\alpha D\left(G\right)+(1-\alpha )A\left(G\right).$$

This paper shows some extremal results on the spectral radius ρ_α(G) of A_α(G). We determine the upper bound on ρ_α(G) if α ∈ (0, 1) and G is a graph with no K_{2, t} (t ≥ 3) minor. We also show that the unique outerplanar graph of order n with maximum ρ_α(G) is the join of a vertex and a path $P_{n-1}$. Moreover, we prove that the unique planar graph of order n with maximum signless Laplacian spectral radius is the join of an edge and a path $P_{n-2}$.

Introduction

In this paper, we only consider undirected simple graphs. Let G be a graph with vertex set V(G) and edge set E(G). The |V(G)| is the order of G. For a vertex v ∈ V(G), we will use d_G(v) and N_G(v) to denote the degree and the neighborhood of v, respectively. If it is clear which graph is being considered, we simply write d(v) and N(v). The minimum degree and maximum degree of G are denoted by δ(G) and $\Delta \left(G\right),$ respectively. For U⊆V(G), G[U] denotes the graph induced by U whose edges are precisely the edges of G with both ends in U. A graph H is called a minor of G if H can be obtained by deleting edges and vertices and by contracting edges. As usual, P_m and K_m denote a path and a complete graph of order m, respectively. K_{k, l} ($k,l\in \mathbb{N}$) denotes the complete bipartite graph with parts of cardinality k and l. The graph K_{1, l} (l ≥ 2) is also called a star. For two disjoint graphs G₁ and G₂, denote by $G_1+G_2$ the disjoint union of G₁ and G₂. By starting with a disjoint union of G₁ and G₂ and adding all possible edges between G₁ and G₂, one obtains the join of G₁ and G₂, denoted by G₁∨G₂.

The adjacency matrix A(G) of a graph G is defined by its entries $a_{ij}=1$ if v_iv_j ∈ E(G) and 0 otherwise, and the degree matrix D(G) of G is the diagonal matrix whose entries are the degrees of the vertices of G. The Laplacian matrix and the signless Laplacian matrix of G are $L\left(G\right)=D\left(G\right)-A\left(G\right)$ and $Q\left(G\right)=D\left(G\right)+A\left(G\right),$ respectively. For any real α ∈ [0, 1], Nikiforov [3] introduced and studied the convex combinations A_α(G) of A(G) and D(G) defined by

$$A_{\alpha }\left(G\right)=\alpha D\left(G\right)+(1-\alpha )A\left(G\right).$$

The spectral radius of A_α(G) is called the α-index of G, written as ρ_α(G).

Let $x={(x_1,x_2,\dots ,x_n)}^T$ be a unit eigenvector to the spectral radius ρ_α(G) of A_α(G). Then the eigenvector equation of A_α(G) is

$${\displaystyle {\left(A_{\alpha }\left(G\right)x\right)}_u=\alpha d_G\left(u\right)x_u+(1-\alpha )\sum _{i\sim u}{x}_i.}$$

Also, we have

$${\displaystyle {\rho }_{\alpha }\left(G\right)=<A_{\alpha }\left(G\right)x,x>=\sum _{ij\in E\left(G\right)}(\alpha x_i^2+2(1-\alpha )x_i{x}_j+\alpha x_j^2).}$$

Suppose that t ≥ 3 and $n\ge t+1$. Let $p=\lfloor (n-1)/t\rfloor$ and $n-1=pt+s$. Define $F_t\left(n\right):=K_1\vee (p{K}_t+K_s).$ With this definition, Nikiforov [4] obtained the following results.

Theorem 1.1

[4] Let $t=3$. If G is a graph of order n > 40000 and with no K_{2, 3} minor, then ρ₀(G) < ρ₀(F₃(n)), unless $G=F_3\left(n\right).$

Theorem 1.2

[4] Let t ≥ 4 and n ≥ 400 · t⁶. If G is a graph of order n and with no K_{2, t} minor, then

$$\begin{array}{c}\hfill {\rho }_0\left(G\right)\le \frac{t-1}{2}+\sqrt{n+\frac{t^2-2t-3}{4}}.\end{array}$$

Equality holds if and only if $n\equiv 1\left(\mathrm{mod}t\right)$ and $G=F_t\left(n\right).$

One of the main goals of this paper is to extend the above theorems to arbitrary α ∈ (0, 1). Moreover, we prove that the unique outerplanar graph of maximum α-index is the join of a vertex and $P_{n-1}$. Also, we show that $K_2\vee P_{n-2}$ is the planar graph with maximum ρ_α(G) when $\alpha \in [0.486,\frac{1}{2}]$.