Computing the Determinant of the Distance Matrix of a Bicyclic Graph

The determinant of the distance matrix of graphs with blocks at most bicyclic

“…Note that θ(l, p, q)-graph is a bicyclic graph, with no pendant edge, whose cycles share at least one edge. In [2], we proved the following results:…”

“…The next theorem gives the determinant of D(G) when G = θ(l, p, q), completing the remaining cases in [2].…”

“…Proof. Items (c) and (d) have been proven in [2] and correspond to Proposition 1 and Proposition 2 respectively. Case (b) can be easily computed.…”

“…In a conference article [2], we presented some advances for the remaining cases; i.e., when the cycles share at least one edge. Besides, we conjectured the formula for the remaining cases.…”

Distance Matrix of Weighted Cactoid-type Digraphs

“…We first recall C(n, m 1 , m 2 , • • • , m r ) is a undirected graph consisting of r cycles sharing a common path of length n. In this section, we compute the determinant of the distance matrix for a class of C(n, m 1 , m 2 , • • • , m r ). In literature, similar problems has been studied for the cases r = 1, 2 (for details see [3,8,9,10,15]) and we have shown that some of their results can be extended for r ≥ 3. We recall some results for the case r = 1, i.e.…”