Some polynomials of flower graphs

Abstract

We define a class of graphs called flower and give some properties of these graphs. Then the explicit expressions of the chromatic polynomial and the flow polynomial is given. Further, we give classes of graphs with the same chromatic and flow polynomials. Mathematics Subject Classification: 05C99

Chromatic and flow polynomials of generalized vertex join graphs and outerplanar graphs

“…, f s , can be obtained by adding 2 to every element of φ(C * S ). Thus, these definitions, along with (11), can be used to compute G i used in the formula above, and thus we can compute P (C S ) and F (C S ) for any generalized vertex join cycle C S .…”

“…5. Decomposing CS (on far left) into simpler graphs as described in (10). Using (1), the graphs in the top row can be further decomposed into the cycles making up their bounded faces, since in these graphs, the faces bordering the contraction share only one edge with the rest of the graph.…”

“…1 if a i = 1, z + 1 if a i is the beginning of a maximal connected substring of zeros of φ(C S ) with length z. (11) Moreover, the sizes of the faces of C S , f 1 , . .…”

“…Whitehead [18,19] characterizes the chromatic polynomials of a class of clique-like graphs called q-trees. Furthermore, Lazuka [7] obtains explicit formulas for the chromatic polynomials of cactus graphs, Gordon [5] studies Tutte polynomials (a generalization of chromatic and flow polynomials) of rooted trees, and Mphako-Banda [10,11] derives formulas for the chromatic, flow, and Tutte polynomials of flower graphs.…”

Defect polynomials and Tutte polynomials of some asymmetric graphsDefect polynomials and Tutte polynomials of some asymmetric graphs

“…Flower graphs form a class of graphs that is highly symmetric. Some classes of flower graphs have attractive simple formulas for the chromatic polynomial; see [8]. In this paper, we study the asymmetric complete flower graph and asymmetric incomplete flower graph.…”