Complementary distance polynomial and energy of join of two graphs

Abstract

For a connected graph G with diameter D, the complementary distance matrix is defined as, CD(G)=[c_ij] in which c_ij=1+D-d_ij if i not equal  j and i otherwise, where d_ij is distance between the vertices v_i and v_j. The CD-polynomial has been studied for the join of two regular graphs when both the graphs are of diameter less than or equal to 2. In the present work, we study CD-polynomial for join of any two graphs and hence construct a pair of CD-equienergetic graphs by joining a regular graph (which is among a pair of CD-equienergetic graphs of same order and degree) with a non-regular graph. Also, CD-eigenvalues for these structures are studied interms of adjacency eigenvalues of G₁ and G₂ when both of them are regular.