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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

E_G(x)=sum_(k=0)^mc_kx^k,
(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).

graphE(x)
complete bipartite graph K_(m,n)sum_(k=0)^(m)(-1)^k(m; k)[(x+1)^(m-k)-1]^n
complete graph K_nsum_(k=0)^(n)(-1)^(n-k)(n; k)(x+1)^((k; 2))
cycle graph C_nx^n+sum_(k=1)^(n-1)n/(n-k)(k-1; n-k-1)x^k
path graph P_nsum_(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.

graphE(x)
book graph S_(n+1) square P_2x^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_nsqrt(x/(x+4))((x+sqrt(x(4+x)))^(n-1)-(x-sqrt(x(4+x)))^(n-1))/(2^(n-1))
star graph S_nx^(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.

graphorderrecurrence
cycle graph C_n2p_n(x)=xp_(n-2)(x)+xp_(n-1)(x)
book graph S_(n+1) square P_23p_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 graph4p_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 graph2p_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_n3p_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_n4p_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_n2p_n(x)=xp_(n-2)(x)+xp_(n-1)(x)
prism graph Y_n4p_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_n1p_n(x)=xp_(n-1)(x)
sunlet graph C_n circledot K_11p_n(x)=x(x+1)p_(n-1)(x)
web graph2p_n(x)=x^3(x+1)^3p_(n-2)(x)+x^2(x+2)(x+1)p_(n-1)(x)
wheel graph W_n4p_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)

See also

Edge Cover, Edge Cover Number, Vertex Cover Polynomial

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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.

Cite this as:

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

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