DENSITY OF THE SHADOW GRAPH OF SOME GRAPHS AND THE JOIN OF GRAPHS

Abstract

Graph density, a classical metric quantifying edge prevalence relative to possible connections, supports the structural analysis of networks across mathematics, computer science, and the social and biological sciences. In this paper, we address the problem of characterizing the density of the shadow graph $D_2(G)$, defined by associating to any undirected connected graph $G$ and whose edges indicate pairs of original edges sharing a common endpoint. Specializing this expression, we obtain explicit density result for the shadow graph of several standard graph families- namely path $P_n$, cycle $C_n$, complete $K_n$, barbell $B_n$, friendship $F r_n$, sunlet $S_n$, banana $B_{n, k}$, gear $G_n$, tadpole $T_{m, n}$, wheel $W_n$, and fan graphs $F_n$. Furthermore, we compute the density of some known graph families as an immediate consequence.

Received: April 22, 2025
Revised: June 2, 2025
Accepted: June 17, 2025