﻿ 一类图的谱
 上海理工大学学报  2019, Vol. 41 Issue (5): 417-421 PDF

Spectra of a Class of Graphs
ZENG Jianyu, HE Changxiang
College of Science, University of Shanghai for Science and Technology, Shanghai 200093, China
Abstract: Let Km is a complete graph of order m. By linking n+1 complete graphs of order m in a fixed way, a complete associated graph ${H_{n,{{K_m}}}}$ of order $\left( {mn + m} \right)$ was obtained. The adjacency eigenvalues, Laplacian eigenvalues and signless Laplacian eigenvalues of ${H_{n,{{K_m}}}}$ were provided according to the relevant conclusions about quotient matrix and rank. So the adjacency spectrum, Laplacian spectrum and signless Laplacian spectrum of ${H_{n,{{K_m}}}}$ were determined. At the same time, based on the study of Brualdi-Solheid spectral radius, the research of this kind of spectral radius problem was extended to the study of Laplacian spectral radius and signless Laplacian spectral radius of graphs. The upper and lower bounds of the radius of adjacency spectrum, Laplacian spectrum and signless Laplacian spectrum of ${{\rm{{\cal H}}}_{n,{K_m}}}$ (the set of complete correlated graphs with points N where $N = m(n + 1)$ ) were provided. And the extremal graphs were described where the radius of adjacency spectrum attained the upper bound and Laplacian spectrum and signless Laplacian spectrum attained the upper or lower bounds.
Key words: spectra     quotient matrix     extremal graph
1 基本概念及背景介绍

$G$ $n$ 阶图，顶点集为 $V\left( G \right) = \left\{ {{v_1},{v_2}, \cdots ,{v_n}} \right\}$ ，边集为 $E\left( G \right)$ ${{A}}\left( G \right)$ 为邻接矩阵，若 ${v_i}$ ${v_j}$ 之间有边相连，则 ${a_{ij}} = 1$ ；否则， ${a_{ij}} = 0$ ${{D}}\left( G \right) = {\rm{diag}} ( d\left( {{v_1}} \right),$ $d\left( {{v_2}} \right), \cdots ,d\left( {{v_n}} \right) )$ 为度对角矩阵， $d\left( {{v_i}} \right)$ 表示顶点 ${v_i}$ 的度。矩阵 ${{L}}\left( G \right) = {{D}}\left( G \right) - {{A}}\left( G \right)$ ${{Q}}\left( G \right) = {{D}}\left( G \right)+ {{A}}\left( G \right)$ 分别称为 $G$ 的拉普拉斯和无符号拉普拉斯矩阵。图 $G$ 的邻接矩阵、拉普拉斯矩阵和无符号拉普拉斯矩阵的特征多项式分别记为 ${\varphi _{{A}}}\left( {G,\lambda } \right)$ , ${\varphi _{{L}}}\left( {G,\mu } \right)$ ${\varphi _{{Q}}}\left( {G,q} \right)$ 。图 $G$ 的邻接矩阵、拉普拉斯矩阵和无符号拉普拉斯矩阵的特征值即分别为 ${\varphi _{{A}}}\left( {G,\lambda } \right)$ , ${\varphi _{{L}}}\left( {G,\mu } \right)$ ${\varphi _{{Q}}}\left( {G,q} \right)$ 的根，分别记为 ${\lambda _i}\left( G \right),\;{\mu _i}\left( G \right)$ ${q_i}\left( G \right)$ $\left( {i = 1,2, \cdots ,n} \right)$ ，它们所构成的多重集合 $\left\{ {{\lambda _1}\left( G \right),} {{\lambda _2}\left( G \right), \cdots ,{\lambda _n}\left( G \right)} \right\}$ , $\left\{ {{\mu _1}\left( G \right),{\mu _2}\left( G \right), \cdots ,{\mu _n}\left( G \right)} \right\}$ $\left\{ {{q_1}\left( G \right),{q_2}\left( G \right), \cdots ,{q_n}\left( G \right)} \right\}$ 分别称为图 $G$ 的邻接谱、拉普拉斯谱和无符号拉普拉斯谱。 $\lambda \left( G \right)$ $\mu \left( G \right)$ $q\left( G \right)$ 分别为邻接、拉普拉斯和无符号拉普拉斯谱的谱半径。

 图 1 书图 ${H_{4,{{K_2}}}}$ Fig. 1 Book graph ${H_{4,{{K_2}}}}$

 图 2 完全关联图 ${H_{3,{{K_3}}}}$ Fig. 2 Complete associated graph ${H_{3,{{K_3}}}}$

 ${{A}}({H_{n,}}_{{K_m}}) \!=\! {\left( \!\!\!\!{\begin{array}{*{20}{c}} {{{A}}\left( {{K_m}} \right)}\!\!&\!\!{{{{E}}_m}}& \cdots &{{{{E}}_m}} \\ {{{{E}}_m}}\!\!&\!\!{{{A}}\left( {{K_m}} \right)}&{}&{} \\ \vdots \!\!&\!\!{}& \ddots &{} \\ {{{{E}}_m}}\!\!&\!\!{}&{}&{{{A}}\left( {{K_m}} \right)} \end{array}} \!\!\!\!\right)_{\left( {mn + m} \right) \times \left( {mn + m} \right)}}$
 ${{D}}\left( {{H_{n,}}_{{K_m}}} \right) \!=\! {\left( \!\!\!\!{\begin{array}{*{20}{c}} {\left({n \!+\! m \!- \!1} \right){{{E}}_m}}\!\!&\!\!{}\!\!&\!\!{}\!\!&{} \\ {}\!\!&\!\!{m{{{E}}_m}}\!\!&\!\!{}&\!\!{} \\ {}\!\!&\!\!{}\!\!&\!\! \ddots &\!\!{} \\ {}\!\!&\!\!{}\!\!&\!\!{}&\!\!{m{{{E}}_m}} \end{array}}\!\!\!\! \right)_{\left( {mn \!+\! m} \right) \times \left( {mn \!+ \!m} \right)}}$

${H_{n,{{K_m}}}}$ 的拉普拉斯和无符号拉普拉斯矩阵分别为

 ${{L}}\left( {H_{n,{{K_m}}}} \right) = {\left( {\begin{array}{*{20}{c}} {\left( {n + m - 1} \right){{{E}}_m} - {{A}}\left( {{K_m}} \right)}&{ - {{{E}}_m}}& \cdots &{ - {{{E}}_m}} \\ { - {{{E}}_m}}&{m{{{E}}_m} - {{A}}\left( {{K_m}} \right)}&{}&{} \\ \vdots &{}& \ddots &{} \\ { - {{{E}}_m}}&{}&{}&{m{{{E}}_m} - {{A}}\left( {{K_m}} \right)} \end{array}} \right)_{\left( {mn + m} \right) \times \left( {mn + m} \right)}}$
 ${{Q}}\left( {H_{n,{{K_m}}}} \right) = {\left( {\begin{array}{*{20}{c}} {\left( {n + m - 1} \right){{{E}}_m} + {{A}}\left( {{K_m}} \right)}&{{{{E}}_m}}& \cdots &{{{{E}}_m}} \\ {{{{E}}_m}}&{m{{{E}}_m} + {{A}}\left( {{K_m}} \right)}&{}&{} \\ \vdots &{}& \ddots &{} \\ {{{{E}}_m}}&{}&{}&{m{{{E}}_m} + {{A}}\left( {{K_m}} \right)} \end{array}} \right)_{\left( {mn + m} \right) \times \left( {mn + m} \right)}}$

${v_{ik}} \in {V_{m + k}}$ ，其中 $i \in \left\{ {1,2, \cdots ,n} \right\}$ $k \in \left\{ {1,2, \cdots ,m} \right\}$ 时， ${v_{ik}}$ 在顶点集 ${V_l}$ 中有且仅有1个邻点，其中 $l = k$ ${v_{ik}}$ 在其他剩余的所有顶点集中都没有邻点。

2 ${H_{n,{{K_m}}}}$ 的各种谱

${{{B}}_1} = - {{E}} - {{A}}\left( {{H_{n,{{K_m}}}}} \right)$ ，则

 \begin{aligned} & R\left( {{{{B}}_1}} \right) = R\left( {\begin{array}{*{20}{c}} {{{{J}}_m}}&{{{{E}}_m}}&{{{{E}}_m}}& \cdots &{{{{E}}_m}} \\ {{{{E}}_m}}&{{{{J}}_m}}&{}&{}&{} \\ {{{{E}}_m}}&{}&{{{{J}}_m}}&{}&{} \\ \vdots &{}&{}& \ddots &{} \\ {{{{E}}_m}}&{}&{}&{}&{{{{J}}_m}} \end{array}} \right) = R\left( {\begin{array}{*{20}{c}} {{{{J}}_m}}&{{{{E}}_m}}&{{{{E}}_m}}& \cdots &{{{{E}}_m}} \\ {{{{E}}_m}}&{{{{J}}_m}}&{}&{}&{} \\ {{O}} &{ - {{{J}}_m}}&{{{{J}}_m}}&{}&{} \\ \vdots & \vdots &{}& \ddots &{} \\ {{O}} &{ - {{{J}}_m}}&{}&{}&{{{{J}}_m}} \end{array}} \right) = \end{aligned}
 $\begin{array}{l} \! R\left( {\begin{array}{*{20}{c}} {{{{J}}_m}}&{{{{E}}_m}}&{{{{E}}_m}}& \cdots &{{{{E}}_m}} \\ {{{{E}}_m}}&{{{{J}}_m}}&{}&{}&{} \\ {\begin{array}{*{20}{c}} {{O}} \\ {{O}} \end{array}}&{\begin{array}{*{20}{c}} { - {{1}}_m^{\rm{T}}} \\ { - {{1}}_{\left( {m - 1} \right) \times m}^{}} \end{array}}&{\begin{array}{*{20}{c}} {{{1}}_m^{\rm{T}}} \\ {{{1}}_{\left( {m - 1} \right) \times m}^{}} \end{array}}&{}&{} \\ \vdots & \vdots &{}& \ddots &{} \\ {\begin{array}{*{20}{c}} {{O}} \\ {{O}} \end{array}}&{\begin{array}{*{20}{c}} { - {{1}}_m^{\rm{T}}} \\ { - {{1}}_{\left( {m - 1} \right) \times m}^{}} \end{array}}&{}&{}&{\begin{array}{*{20}{c}} {{{1}}_m^{\rm{T}}} \\ {{{1}}_{\left( {m - 1} \right) \times m}^{}} \end{array}} \end{array}}\! \!\!\right) \!=\! R\left(\!\!\! {\begin{array}{*{20}{c}} {{{{J}}_m}}&{{{{E}}_m}}&{{{{E}}_m}}& \cdots &{{{{E}}_m}} \\ {{{{E}}_m}}&{{{{J}}_m}}&{}&{}&{} \\ {\begin{array}{*{20}{c}} {{O}} \\ {{O}} \end{array}}&{\begin{array}{*{20}{c}} { - {{1}}_m^{\rm{T}}} \\ {{{{O}}_{\left( {m - 1} \right) \times m}}} \end{array}}&{\begin{array}{*{20}{c}} {{{1}}_m^{\rm{T}}} \\ {{{{O}}_{\left( {m - 1} \right) \times m}}} \end{array}}&{}&{} \\ \vdots & \vdots &{}& \ddots &{} \\ {\begin{array}{*{20}{c}} {{O}} \\ {{O}} \end{array}}&{\begin{array}{*{20}{c}} { - {{1}}_m^{\rm{T}}} \\ {{{{O}}_{\left( {m - 1} \right) \times m}}} \end{array}}&{}&{}&{\begin{array}{*{20}{c}} {{{1}}_m^{\rm{T}}} \\ {{{{O}}_{\left( {m - 1} \right) \times m}}} \end{array}} \end{array}} \right) =\\ \qquad \left( {n + 1} \right)m - (m - 1)(n - 1) \\ \end{array}$

${{{B}}_2} = \left( {m - 1} \right){{E}} - {{A}}\left( {{H_{n,{{K_m}}}}} \right)$ ，则

 \begin{aligned} &\quad R\left( {{{{B}}_2}} \right) = R\left( {\begin{array}{*{20}{c}} {\left( {1 - m} \right){{{E}}_m} + {{A}}\left( {{K_m}} \right)}&{{{{E}}_m}}&{{{{E}}_m}}& \cdots &{{{{E}}_m}} \\ {{{{E}}_m}}&{\left( {1 - m} \right){{{E}}_m} + {{A}}\left( {{K_m}} \right)}&{}&{}&{} \\ {{{{E}}_m}}&{}&{\left( {1 - m} \right){{{E}}_m} + {{A}}\left( {{K_m}} \right)}&{}&{} \\ \vdots &{}&{}& \ddots &{} \\ {{{{E}}_m}}&{}&{}&{}&{\left( {1 - m} \right){{{E}}_m} + {{A}}\left( {{K_m}} \right)} \end{array}} \right) =\\ &\qquad R\left( {\begin{array}{*{20}{c}} {\left( {1 - m} \right){{{E}}_m} + {{A}}\left( {{K_m}} \right)}&{{{{E}}_m}}&{{{{E}}_m}}& \cdots &{{{{E}}_m}} \\ {{{{E}}_m}}&{\left( {1 - m} \right){{{E}}_m} + {{A}}\left( {{K_m}} \right)}&{}&{}&{} \\ {{O}} &{\left( {m - 1} \right){{{E}}_m} - {{A}}\left( {{K_m}} \right)}&{\left( {1 - m} \right){{{E}}_m} + {{A}}\left( {{K_m}} \right)}&{}&{} \\ \vdots & \vdots &{}& \ddots &{} \\ {{O}} &{\left( {m - 1} \right){{{E}}_m} - {{A}}\left( {{K_m}} \right)}&{}&{}&{\left( {1 - m} \right){{{E}}_m} + {{A}}\left( {{K_m}} \right)} \end{array}} \right) \end{aligned}

${{C}} = \left( {1 - m} \right){{{E}}_m} + {{A}}\left( {{K_m}} \right)$ ，因为

 \begin{aligned} & R\left( {{C}} \right)\! =\! R\left( \!\!\!{\begin{array}{*{20}{c}} {1 \!-\! m}&1& \cdots &1 \\ 1&{1 \!-\! m}& \cdots &1 \\ \vdots & \vdots &{}& \vdots \\ 1&1& \cdots &{1 - m} \end{array}} \!\!\!\right) = R\left( \!\!\!\!{\begin{array}{*{20}{c}} {1 \!-\! m}&1& \cdots &1 \\ m&{ - m}& \cdots &0 \\ \vdots & \vdots &{}& \vdots \\ m&0& \cdots &{ - m} \end{array}}\!\!\! \right) = R\left( \!\!\!{\begin{array}{*{20}{c}} {1 \!-\! m}&1& \cdots &1 \\ 1&{ - 1}& \cdots &0 \\ \vdots & \vdots &{}& \vdots \\ 1&0& \cdots &{ - 1} \end{array}}\!\!\! \right) = R\left( \!\!\!{\begin{array}{*{20}{c}} 0&0& \cdots &0 \\ 1&{ - 1}& \cdots &0 \\ \vdots & \vdots &{}& \vdots \\ 1&0& \cdots &{ - 1} \end{array}} \!\!\!\right) =\\ & \quad m - 1 \end{aligned}

${{{B}}_3} = \left( { - {\rm{1 + }}\sqrt n } \right){{E}} - {{{S}}_{{{{A}}_m}}}$ ，则

 \begin{aligned} &\quad R\left( {{B}} \right) \!= \!R\left(\!\!\!\! {\begin{array}{*{20}{c}} {(1 \!-\! \sqrt n ){{{E}}_m} + {{A}}({K_m})}&{n{{{E}}_m}}\\ {{{{E}}_m}}&{(1 \!-\! \sqrt n ){{{E}}_m} + {{A}}({K_m})} \end{array}} \!\!\!\!\right)=\\ &\qquad R\left( \!\!\!\!{\begin{array}{*{20}{c}} {1 - \sqrt n }\!\!\!&\!\!\!1\!\!\!&\!\!\! \cdots \!\!\!&\!\!\!1\!\!\!&\!\!\!n\!\!\!&\!\!\!0\!\!\!&\!\!\! \cdots \!\!\!&\!\!\!0\\ 1\!\!\!&\!\!\!{1 \!-\! \sqrt n }\!\!\!&\!\!\! \cdots \!\!\!&\!\!\!1\!\!\!&\!\!\!0\!\!\!&\!\!\!n\!\!\!&\!\!\! \cdots \!\!\!&\!\!\!0\\ \vdots \!\!\!&\!\!\! \vdots \!\!\!&\!\!\!{}\!\!\!&\!\!\! \vdots \!\!\!&\!\!\! \vdots \!\!\!&\!\!\! \vdots \!\!\!&\!\!\!{}\!\!\!&\!\!\! \vdots \\ 1\!\!\!&\!\!\!1\!\!\!&\!\!\! \cdots \!\!\!&\!\!\!{1 \!-\! \sqrt n }\!\!\!&\!\!\!0\!\!\!&\!\!\!0\!\!\!&\!\!\! \cdots \!\!\!&\!\!\!n\\ 1\!\!\!&\!\!\!0\!\!\!&\!\!\! \cdots \!\!\!&\!\!\!0\!\!\!&\!\!\!{1 - \sqrt n }\!\!\!&\!\!\!0\!\!\!&\!\!\! \cdots \!\!\!&\!\!\!0\\ 0\!\!\!&\!\!\!1\!\!\!&\!\!\! \cdots \!\!\!&\!\!\!0\!\!\!&\!\!\!0\!\!\!&\!\!\!{1 - \sqrt n }\!\!\!&\!\!\! \cdots \!\!\!&\!\!\!0\\ \vdots \!\!\!&\!\!\! \vdots \!\!\!&\!\!\!{}\!\!\!&\!\!\! \vdots \!\!\!&\!\!\! \vdots \!\!\!&\!\!\! \vdots \!\!\!&\!\!\!{}\!\!\!&\!\!\! \vdots \\ 0\!\!\!&\!\!\!0\!\!\!&\!\!\!0\!\!\!&\!\!\!1\!\!\!&\!\!\!0\!\!\!&\!\!\!0\!\!\!&\!\!\! \cdots \!\!\!&\!\!\!{1 \!- \!\sqrt n } \end{array}} \right) \!=\! R\left( {\begin{array}{*{20}{c}} {1 - \sqrt n }\!\!\!&\!\!\!1\!\!\!&\!\!\! \cdots \!\!\!&\!\!\!1\!\!\!&\!\!\!n\!\!\!&\!\!\!0\!\!\!&\!\!\! \cdots \!\!\!&\!\!\!0\\ {\sqrt n }\!\!\!&\!\!\!{ - \sqrt n }\!\!\!&\!\!\! \cdots \!\!\!&\!\!\!0\!\!\!&\!\!\!{ - n}\!\!\!&\!\!\!n\!\!\!&\!\!\! \cdots \!\!\!&\!\!\!0\\ \vdots \!\!\!&\!\!\! \vdots \!\!\!&\!\!\!{}\!\!\!&\!\!\! \vdots \!\!\!&\!\!\! \vdots \!\!\!&\!\!\! \vdots \!\!\!&\!\!\!{}\!\!\!&\!\!\! \vdots \\ {\sqrt n }\!\!\!&\!\!\!0\!\!\!&\!\!\! \cdots \!\!\!&\!\!\!{ - \sqrt n }\!\!\!&\!\!\!{ - n}\!\!\!&\!\!\!0\!\!\!&\!\!\! \cdots \!\!\!&\!\!\!n\\ 1\!\!\!&\!\!\!0\!\!\!&\!\!\! \cdots \!\!\!&\!\!\!0\!\!\!&\!\!\!{1 - \sqrt n }\!\!\!&\!\!\!0\!\!\!&\!\!\! \cdots \!\!\!&\!\!\!0\\ { - 1}\!\!\!&\!\!\!1\!\!\!&\!\!\! \cdots \!\!\!&\!\!\!0\!\!\!&\!\!\!{\sqrt n }\!\!\!&\!\!\!{ - \sqrt n }\!\!\!&\!\!\! \cdots \!\!\!&\!\!\!0\\ \vdots \!\!\!&\!\!\! \vdots \!\!\!&\!\!\!{}\!\!\!&\!\!\! \vdots \!\!\!&\!\!\! \vdots \!\!\!&\!\!\! \vdots \!\!\!&\!\!\!{}\!\!\!&\!\!\! \vdots \\ { - 1}\!\!\!&\!\!\!0\!\!\!&\!\!\!0\!\!\!&\!\!\!1\!\!\!&\!\!\!{\sqrt n }\!\!\!&\!\!\!0\!\!\!&\!\!\! \cdots \!\!\!&\!\!\!{ - \sqrt n } \end{array}}\!\!\!\! \right)= \end{aligned}
 $\begin{array}{l} R\left( {\begin{array}{*{20}{c}} {1 \!-\! \sqrt n }&1& \cdots &1&n&0& \cdots &0\\ {\sqrt n }&{ - \sqrt n }& \cdots &0&{ - n}&n& \cdots &0\\ \vdots & \vdots &{}& \vdots & \vdots & \vdots &{}& \vdots \\ {\sqrt n }&0& \cdots &{ - \sqrt n }&{ - n}&0& \cdots &n\\ 1&0& \cdots &0&{1 \!-\! \sqrt n }&1& \cdots &1\\ {\sqrt n }&{ - \sqrt n }& \cdots &0&{ - n}&n& \cdots &0\\ \vdots & \vdots &{}& \vdots & \vdots & \vdots &{}& \vdots \\ {\sqrt n }&0&0&{ - \sqrt n }&{ - n}&0& \cdots &n \end{array}} \right) \!=\! R\left( {\begin{array}{*{20}{c}} {1 \!-\! \sqrt n }&1& \cdots &1&n&0& \cdots &0\\ {\sqrt n }&{ - \sqrt n }& \cdots &0&{ - n}&n& \cdots &0\\ \vdots & \vdots &{}& \vdots & \vdots & \vdots &{}& \vdots \\ {\sqrt n }&0& \cdots &{ - \sqrt n }&{ - n}&0& \cdots &n\\ 1&0& \cdots &0&{1 \!-\! \sqrt n }&1& \cdots &1\\ 0&0& \cdots &0&0&0& \cdots &0\\ \vdots & \vdots &{}& \vdots & \vdots & \vdots &{}& \vdots \\ 0&0&0&0&0&0& \cdots &0 \end{array}} \right)\!=\\ \left( {n + 1} \right)m - \left( {m - 1} \right) \end{array}$

 $\begin{split} &\quad S({{L}}) = \left\{ {0,{1^{(n - 1)}},{m^{(m - 1)}},{{(m + 1)}^{[(m - 1)(n - 1)]}},}\right.\\ &\qquad \left.{ n + 1,{{[n + (m + 1)]}^{(m - 1)}}} \right\} \end{split}$
 $\begin{split} & \!\!\!\!\!\!S({{Q}}) = \left\{ {{{(m - 2)}^{(m - 1)}},{{(m - 1)}^{(m - 1)(n - 1)}},}\right.\\ & \!\!\!\!\!\!\quad{2(m - 1),{{(2m - 1)}^{(n - 1)}},}\\ &\!\!\!\!\!\!\quad \left.{n + (2m - 1),{{[n + (m - 1)]}^{(m - 1)}}} \right\} \end{split}$

3 ${H_{n,{{K_m}}}}$ 的谱半径

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