The Distance Laplacian Spectral Radius of Clique Trees
Xiao Zhang, Jiajia Zhou
Abstract
The distance Laplacian matrix of a connected graph <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" id="M1"> <a:mi>G</a:mi> </a:math> is defined as <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" id="M2"> <c:mi mathvariant="normal">ℒ</c:mi> <c:mfenced open="(" close=")" separators="|"> <c:mrow> <c:mi>G</c:mi> </c:mrow> </c:mfenced> <c:mo>=</c:mo> <c:mtext>Tr</c:mtext> <c:mfenced open="(" close=")" separators="|"> <c:mrow> <c:mi>G</c:mi> </c:mrow> </c:mfenced> <c:mo>−</c:mo> <c:mi>D</c:mi> <c:mfenced open="(" close=")" separators="|"> <c:mrow> <c:mi>G</c:mi> </c:mrow> </c:mfenced> </c:math> , where <o:math xmlns:o="http://www.w3.org/1998/Math/MathML" id="M3"> <o:mi>D</o:mi> <o:mfenced open="(" close=")" separators="|"> <o:mrow> <o:mi>G</o:mi> </o:mrow> </o:mfenced> </o:math> is the distance matrix of <t:math xmlns:t="http://www.w3.org/1998/Math/MathML" id="M4"> <t:mi>G</t:mi> </t:math> and <v:math xmlns:v="http://www.w3.org/1998/Math/MathML" id="M5"> <v:mtext>Tr</v:mtext> <v:mfenced open="(" close=")" separators="|"> <v:mrow> <v:mi>G</v:mi> </v:mrow> </v:mfenced> </v:math> is the diagonal matrix of vertex transmissions of <ab:math xmlns:ab="http://www.w3.org/1998/Math/MathML" id="M6"> <ab:mi>G</ab:mi> </ab:math> . The largest eigenvalue of <cb:math xmlns:cb="http://www.w3.org/1998/Math/MathML" id="M7"> <cb:mi mathvariant="normal">ℒ</cb:mi> <cb:mfenced open="(" close=")" separators="|"> <cb:mrow> <cb:mi>G</cb:mi> </cb:mrow> </cb:mfenced> </cb:math> is called the distance Laplacian spectral radius of <ib:math xmlns:ib="http://www.w3.org/1998/Math/MathML" id="M8"> <ib:mi>G</ib:mi> </ib:math> . In this paper, we determine the graphs with maximum and minimum distance Laplacian spectral radius among all clique trees with <kb:math xmlns:kb="http://www.w3.org/1998/Math/MathML" id="M9"> <kb:mi>n</kb:mi> </kb:math> vertices and <mb:math xmlns:mb="http://www.w3.org/1998/Math/MathML" id="M10"> <mb:mi>k</mb:mi> </mb:math> cliques. Moreover, we obtain <ob:math xmlns:ob="http://www.w3.org/1998/Math/MathML" id="M11"> <ob:mi>n</ob:mi> </ob:math> vertices and <qb:math xmlns:qb="http://www.w3.org/1998/Math/MathML" id="M12"> <qb:mi>k</qb:mi> </qb:math> cliques.