Edge Cover Polynomial

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Let c_k be the number of edge covers of a graph G of size k. Then the edge cover polynomial E_G(x) is defined by

(1)

where m is the edge count of G (Akban and Oboudi 2013).

Cycle graphs and complete bipartite graphs are determined by their edge cover polynomials (Akban and Oboudi 2013).

The edge cover polynomial is multiplicative over graph components, so for a graph G having connected components G_1, G_2, ..., the edge cover polynomial of G itself is given by

E_G=E_(G_1)E_(G_2)....

(2)

The edge cover polynomial satisfies

E_G(-1)=(-1)^nI_G(-1),

(3)

where n=|G| is the vertex count of a graph G and I_G(x) is its independence polynomial (Akban and Oboudi 2013).

The following table summarizes sums for the edge cover polynomials of some common classes of graphs (Akban and Oboudi 2013).

graph E(x)

complete bipartite graph K_(m,n) [画像:sum_(k=0)^(m)(-1)^k(m; k)[(x+1)^(m-k)-1]^n]

complete graph K_n [画像:sum_(k=0)^(n)(-1)^(n-k)(n; k)(x+1)^((k; 2))]

cycle graph C_n [画像:x^n+sum_(k=1)^(n-1)n/(n-k)(k-1; n-k-1)x^k]

path graph P_n [画像:sum_(k=1)^(n-1)(k-1; n-k-1)x^k]

The following table summarizes closed forms for the edge cover polynomials of some common classes of graphs.

graph E(x)

book graph S_(n+1) square P_2 x^n-2[x(1+x)]^n+(1+x)[x(1+x(3+x))]^n

cycle graph C_n ((x-sqrt(x(4+x)))^n+(x+sqrt(x(4+x)))^n)/(2^n)

helm graph -[x(1+x)]^n+[x(1+x)^2]^n

path graph P_n sqrt(x/(x+4))((x+sqrt(x(4+x)))^(n-1)-(x-sqrt(x(4+x)))^(n-1))/(2^(n-1))

star graph S_n x^(n-1)

sunlet graph C_n circledot K_1 [x(1+x)]^n

The following table summarizes the recurrence relations for edge cover polynomials for some simple classes of graphs.

graph order recurrence

cycle graph C_n 2 p_n(x)=xp_(n-2)(x)+xp_(n-1)(x)

book graph S_(n+1) square P_2 3 p_n(x)=(x+1)(x^2+3x+1)x^3p_(n-3)(x)-(x^3+5x^2+8x+3)x^2p_(n-2)(x)+(x+1)(x+3)xp_(n-1)(x)

gear graph 4 p_n(x)=-(x+1)x^4p_(n-4)(x)+2(x+1)(x+2)x^3p_(n-3)(x)-(x+2)(x^2+3x+3)x^2p_(n-2)(x)+(x+2)^2xp_(n-1)(x)

helm graph 2 p_n(x)=x(x+1)(x+2)p_(n-1)(x)-x^2(x+1)^3p_(n-2)(x)

ladder graph P_2 square P_n 3 p_n(x)=-(x+1)x^3p_(n-3)(x)+(x+2)x^3p_(n-2)(x)+(x+1)(x+2)xp_(n-1)(x)

Möbius ladder M_n 4 p_n(x)=-(x+1)x^4p_(n-4)(x)+(x+2)(2x+1)x^2p_(n-2)(x)+(x^2+3x+1)xp_(n-1)(x)+(x^2+x-1)x^3p_(n-3)(x)

path graph P_n 2 p_n(x)=xp_(n-2)(x)+xp_(n-1)(x)

prism graph Y_n 4 p_n(x)=-(x+1)x^4p_(n-4)(x)+(x+2)(2x+1)x^2p_(n-2)(x)+(x^2+3x+1)xp_(n-1)(x)+(x^2+x-1)x^3p_(n-3)(x)

star graph S_n 1 p_n(x)=xp_(n-1)(x)

sunlet graph C_n circledot K_1 1 p_n(x)=x(x+1)p_(n-1)(x)

web graph 2 p_n(x)=x^3(x+1)^3p_(n-2)(x)+x^2(x+2)(x+1)p_(n-1)(x)

wheel graph W_n 4 p_n(x)=-(x+1)x^2p_(n-4)(x)-(2x+3)x^2p_(n-3)(x)-(x-1)(x+2)xp_(n-2)(x)+(x+3)xp_(n-1)(x)

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References

Akban, S. and Oboudi, M. R. "On the Edge Cover Polynomial of a Graph." Europ. J. Combin. 34, 297-321, 2013.

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Edge Cover Polynomial

Cite this as:

Weisstein, Eric W. "Edge Cover Polynomial." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/EdgeCoverPolynomial.html

Edge Cover Polynomial

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