Convex Subgraph Polynomials of the Join and the Composition of Graphs

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1 International Journal of Mathematical Analysis Vol 10, 2016, no 11, HIKARI Ltd, wwwm-hikaricom Convex Subgraph Polynomials of the Join and the Composition of Graphs Ladznar S Laja 1 Department of Mathematics and Sciences, College of Arts and Sciences Mindanao State University - Tawi-Tawi Tawi-Tawi College of Technology and Oceanography Sanga-Sanga, 7500 Bongao, Tawi-Tawi, Philippines Rosalio G Artes, Jr Department of Mathematics and Statistics, College of Science and Mathematics Mindanao State University - Iligan Institute of Technology Andres Bonifacio Avenue, Tibanga 9200 Iligan City, Philippines Copyright c 2015 Ladznar S Laja and Rosalio G Artes, Jr This article is distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited Abstract This paper characterizes the convex subgraphs of the join and the composition of graphs Moreover, we establish the convex subgraph polynomials of graphs resulting from the two graph operations Mathematics Subject Classification: 12D10 Keywords: convex subgraph, convex subgraph polynomial, join of graphs, composition of graphs 1 Research is partially funded by the Commission on Higher Education (CHED), Philippines under the Faculty Development Program

2 516 Ladznar S Laja and Rosalio G Artes, Jr 1 Introduction There are a number of graph polynomials that have been widely studied Chromatic polynomials count the number of proper colourings of a graph [6, 17, 19, 31] Matching polynomials enumerate matching [26] Independence polynomials are generating polynomials for the number of independent sets of each cardinality [27] One of the most general approaches to graph polynomials was proposed by Farrel in 1979 in his theory of F -polynomials of a graph According to Farrel [21], any such polynomial corresponds to a strictly prescribed family of connected subgraphs of the respective graph For the matching polynomial of a graph G, this family consist of all edges of G, for the independence polynomial of G, this family includes all the stable sets of G J I Brown et al [8], examined the effects of various graph operations on neighborhood polynomials, which are generating functions for the number of faces of each cardinality in the neighborhood complex of a graph They provide explicit polynomials for hypercubes, for graphs not containing four-cycle and for graphs resulting from joins and Cartesian products FM Dong et al [20], had determined the vertex-cover polynomial of the path, cycle, wheel, and complete bipartite graph Moreover, they developed a method to calculate the vertex-cover polynomial of a graph Motivated by a problem in biological systematics, they consider a mapping f from {1, 2,, m} into the vertex set of a graph, subject to f 1 (u) f 1 (v) for every edge xy in G They showed that the number of such mappings can be determined from the vertex-cover polynomial Saieed Akbari et al [2] introduced the edge cover polynomial They showed that if E(G, x) = E(H, x), then the degree sequence of G and H are the same They showed that cycles and complete bipartite graphs are determined by their edge cover polynomials Also they determined all graphs G for which E(G, x) = E(P n, x) AVijayan [29] introduced a total edge fixed geodominating sets and polynomials of graphs G t (G, x) They obtained some properties of G t (G, x) and its coefficients They also compute polynomials for complete graph, bipartite graph and the corona of any graph G with complete graph K 1 of order 1 Ali et al [3] obtained the Wiener polynomial W n (G, x) for some special graphs including paths and cycle graphs Moreover, for vertex-disjoint connected graphs G 1 and G 2 formulas for Wiener polynomials of Steiner n- distance of compound graphs are also obtained in terms of those polynomials for G 1 and G 2 Given a graph G of order n, the clique polynomial of G is defined by n ω(g, x) = ω k (G)x k, where ω 0 (G) = 1 and ω k (G) is the number of k-cliques k=0

3 Convex subgraph polynomials of the join and the composition of graphs 517 (complete subgraph of order k) of G The following are some known results in clique and independence polynomials of graphs Theorem 11 [22, 27] Let G and H be two vertex-disjoint graphs, ie, V (G) V (H) = Then (i) ω(g, x) = I(G, x) (ii) ω(g, x) + ω(g, x) = I(G, x) + I(G, x) (iii) ω(g H, x) = ω(g, x) + ω(h, x) 1 (iv) I(G 1 G 2, x) = I(G 1, x)i(g 2, x) (v) ω(g + H, x) = ω(g, x)ω(h, x) (vi) I(G + H) = I(G, x) + I(H, x) 1 Theorem 12 [24] Let G 1 and G 2 be two vertex-disjoint graphs with V (G 1 ) = n 1 and V (G 2 ) = n 2, then ω (G 1 G 2, x) = n 2 ω(g 1, x) + n 1 ω (G 2, x) (n 1 + n 2 + n 1 n 2 x) + 1 Theorem 13 [27] I(P n, x) = F n+1 (x) and I(C n, x) = F n 1 (x)+2xf n 2 (x), where F n (x), for all n 0, are the so-called Fibonacci polynomials, that is, the polynomial defined recursively by F 0 (x) = 1, F 1 (x) = 1,F n (x) = F n 1 (x)+ xf n 2 (x) Corollary 14 [27] I(P n, x) = Theorem 15 [9] I(C n, x) = Theorem 16 [30] I(C n, x) = n+1 2 j=0 n 2 n n j j=0 n 2 s=1 ( n + 1 j j ( n j j [ 1 + 4x cos ) x j ) x j 2 (2s 1)π 2n The desire to make a similar study on the work of the authors in [2, 4, 5, 8] on graphs polynomials motivated the researchers to try this investigation Our investigation uses mainly the concept of convex subgraphs The next section is of great significance for deeper understanding and importance of convexity in graphs ]

4 518 Ladznar S Laja and Rosalio G Artes, Jr 2 Convexity in Graphs The concept of convexity which was mainly defined and studied in R n in the pioneering works of Newton, Minkowski and others as described in [7], now finds a place in many other mathematical structures such as vector spaces, lattices, metric spaces and graphs This concept is important in several applied mathematical areas like optimization, approximation theory, game theory and probability theory [18] It is natural that the concept of convexity could be introduced in graphs also, via intrinsic metric The concept of convexity in graphs is discussed in the book by Buckley and Harary [10] This concept was also investigated by Harary and Nieminen [23] Sampathkumar [28] also studied convexity in graphs where he obtained characterizations of block, cut point and a tree in terms of convex sets Let G be a connected graph For two vertices u and v of a connected graph G, I G [u, v] is the set of all vertices in any u-v geodesic A subset S of V (G) is convex if for every two vertices u, v S, the vertex set of every u-v geodesic is contained in S Equivalently, S is convex if for every two vertices u, v S the closed interval I G [u, v] S The convexity number of G, denoted by con(g), is the maximum cardinality of a proper convex set in G, that is con(g) = max{ S : S is convex in G and S V (G)} Since the empty set is convex, con(k 1 ) = 0, where K 1 is the complete graph with one vertex Chartrand, Fink, and Zhang [14] characterized oriented graphs having convexity number n 1 They also presented some realization theorem in the convexity number of connected oriented graphs and determined the lower orientable convexity number of complete graphs, complete bipartite graphs and outer planar graphs Chartrand, etal [16] stated the following observation: If a connected graph G of order n has end-vertex u, then V (G) \ {u} is convex In paticular, if the minimum degree of G is 1, then con(g) = n 1 In fact, they also characterized a connected graph of order n for which con(g) = n 1 By definition, the coefficients of the convex subgraphs polynomial counts the number of convex subgraphs of specific order in a graph It seems reasonable, therefore, to expect this study to rely much on the available characterization of convex sets in a graph Foremost among these results on the subject will be the work of Canoy, et al [12, 11] which, among many results, gave characterizations of convex sets of two graphs under the following binary operations: join, cartesian product, and composition Given two graphs G and H, Canoy and Garces in [12] characterized the convex sets of the graphs G + H, G[H] and G H As a tool in characterizing the convex sets in G+H, they defined and used the notion of non-connectivity

5 Convex subgraph polynomials of the join and the composition of graphs 519 For connected graphs G and H, they characterized the proper convex subsets of V (G + H) in relation to some sets that induce complete subgraphs of G and H or to some non-connecting sets in G They also proved that the proper convex subsets of G[H] are those that induce complete subgraphs of G[H] Also, they showed that the proper convex subsets of V (G H) are actually the cartesian product of some proper convex subsets of V (G) and V (H) These characterizations eventually paved the way for the determination of the convexity numbers of the graphs G + H, G[H], and G H Byung Kee Kim [25], showed that every pair k, n of integers with 2 k n 1 is realizable as the convexity number and order, respectively, of some connected triangle-free graph G, that is, G does not contain a subgraph K 3 and give a lower bound for the convexity number of k-regular graphs of order n with n > k + 1 They also provided an examples of polyconvex and non polyconvex graphs The following are some known results on convexity in graphs The first theorem appeared in [16], as cited in [15], means that connected graph containing extreme vertices are precisely those having convexity numbers n 1 A vertex v in a connected graph G is an extreme vertex or a complete vertex if the neighborhood N G (v) of v induces a complete subgraph of G The set of extreme vertices of G is denoted by Ext(G) Theorem 21 [16] Let G be a connected graph of order n Then con(g) = n 1 if and only if G has an extreme vertex Theorem 22 [12] Let G and H be connected non complete graphs and C V (G[H]) Then C is convex in G[H] if and only if C induces a complete subgraph of G[H] Theorem 23 [12] Let G and H be a connected graphs Then C V (G H) is convex in G H if and only if C = C G C H, where C G = {u : (u, v) C for some v V (H)} and C H = {v : (u, v) C for some u V (G)} are convex in G and H, respectively Theorem 24 [11] Let G and H be noncomplete graphs Then a proper subset C = S 1 S 2 of V (G + H), where S 1 V (G) and S 2 V (H), is convex set in G + H if and only if S 1 and S 2 induce complete subgraphs of G and H, respectively, where it may occur that S 1 = or S 2 =

6 520 Ladznar S Laja and Rosalio G Artes, Jr Theorem 25 [13] Let G and H be graph A subset of V (G [H]) induces a clique of G[H] if and only if C = ({s} T s ), where S is a clique of G s S and T s is a clique of H for each s S We formally define the convex subgraph polynomial in the following: Let G be a connected graph of order n A convex subgraph of G is a subgraph of G induced by a convex subset of V (G) For each i {0, 1, 2,, n, }, let c i (G) be the number of convex subgraphs of G of order i The convex subgraph polynomial of G, denoted by C(G, x), is the polynomial given by n C(G, x) = c i (G)x i i=0 3 Join of Graphs First of all, we formally define the join of two vextex-disjoint graphs Definition 31 The join G + H of two graphs G and H is the graph with vertex set V (G + H) = V (G) V (H) and edge set E(G + H) = E(G) E(H) {uv : u V (G) and v V (H)} The fan F n of order n + 1 is the graph P n + K 1 The wheel W n of order n + 1 is the graph C n + K 1 Illustration 32 The join C 3 + P 3 of the graphs C 3 and P a a b 2 b 2 c c 3 3 The next result shows that for some graph G, C(G, x) and ω(g, x) are closely related

7 Convex subgraph polynomials of the join and the composition of graphs 521 Proposition 33 Let G be a noncomplete connected graph of order n 3 If every nonempty proper convex set in G induces a complete subgraph of G, then C(G, x) = ω(g, x) + x n Proof : Here con(g)=ω(g) and for each i, 0 i con(g), c i (G)=ω i (G) Also, for every ω(g) < i < n, every subgraph of G of order i is not convex, and so c i (G) = 0 Thus, C(G, x) = 1 + nx + c 2 (G)x 2 + c 3 (G)x c con(g) (G)x con(g) + x n = 1 + ω 1 (G)x + ω 2 (G)x 2 + ω 3 (G)x ω ω(g) (G)x ω(g) + x n = ω(g, x) + x n The next results depend on Theorem 24 [11] which gives the characterization of convex sets in the join of graphs Remark 34 Let G 1 and G 2 be non-complete graphs, and H a proper subgraph of G 1 + G 2 Then H is a convex subgraph of G 1 + G 2 if and only if one of the following holds: (i) H is a complete subgraph of G 1 ; (ii) H is a complete subgraph of G 2 ; (iii) H = H 1 + H 2, where H 1 and H 2 are complete subgraphs of G 1 and G 2, respectively Theorem 35 Let G 1 and G 2 be noncomplete connected graphs of orders n 1 and n 2, respectively Then C(G 1 + G 2, x) = ω(g 1, x)ω(g 2, x) + x n 1+n 2 Proof : Let S V (G 1 + G 2 ) be a nonempty proper convex set in G 1 + G 2 By Remark 34, one of the following holds: (i) S is a complete subgraph of G 1 ; (ii) S is a complete subgraph of G 2 ; (iii) S = H 1 + H 2, where H 1 and H 2 are complete subgraphs of G 1 and G 2, respectively In any case, S is a complete subgraph of G 1 + G 2 By Proposition 33 and Theorem 11, C(G 1 + G 2, x) = ω(g 1, x)ω(g 2, x) + x n 1+n 2

8 522 Ladznar S Laja and Rosalio G Artes, Jr Remark 36 Note that for a graph described in Proposition 33, so that by Theorem 35, ω(g, x) = C(G, x) x n C(G 1 + G 2, x) = [C(G 1, x) x n 1 ] [C(G 2, x) x n 2 ] + x n 1+n 2 for every graphs G 1 and G 2 satisfying the conditions in Proposition 33 Corollary 37 i F or m, n 3, C(T m + T n, x) = 1 + (m + n)x + (m + n + mn 2)x 2 + (2mn m n)x 3 + (m 1)(n 1)x 4 + x m+n ii F or m 3 and n 4, C(T m + C n, x) = 1 + (m + n)x + (m + n + mn 1)x 2 + (2mn n)x 3 + (m 1)nx 4 + x m+n iii F or m, n 4, C(C m + C n, x) = 1 + (m + n)x + (m + n + mn)x 2 + (2mn)x 3 + mnx 4 + x m+n iv F or m, n 3, C(T m + F n, x) = 1 + (m + n + 1)x + (2m + 2n + mn 2)x 2 + (3mn 2)x 3 + ( m 2n + 3mn + 1)x 4 + (m 1)(n 1)x 5 + x m+n+1 v F or m 4 and n 3, C(C m + F n, x) = 1 + (m + n + 1)x + (2m + n + 2mn 1)x 2 + (n + 3mn 1)x 3 + ( 2m + 3mn)x 4 + m(n 1)x 5 + x m+n+1 vi F or m 3 and n 4, C(T m + W n, x) = 1 + (m + n + 1)x + (2m + 2n + mn 1)x 2 + (m + 3mn 1)x 3 + (3m 2)nx 4 + (m 1)nx 5 + x m+n+1 vii F or m, n 4, C(C m + W n, x) = 1 + (m + n + 1)x + (m + 2n + mn + 1)x 2 + (m + n + 3mn)x 3 + (3mn)x 4 + mnx 5 + x m+n+1

9 Convex subgraph polynomials of the join and the composition of graphs 523 viii F or m, n 3, C(F m + F n, x) = 1 + (m + n + 2)x + (3m + 3n + mn 1)x 2 ix F or m 3 and n 4, + (2m + 2n + 4mn 4)x 3 + ( 2m 2n + 6mn 1)x 4 + ( 3m 3n + 4mn + 2)x 5 + (m 1)(n 1)x 6 + x m+n+2 C(F m + W n, x) = 1 + (m + n + 2)x + (3m + 3n + mn)x 2 x F or m, n 4, + (3m + n + 3mn 2)x 3 + (m 2n + 4mn 1)x 4 + ( 3n + 4mn)x 5 + (m 1)nx 6 + x m+n+2 C(W m + W n, x) = 1 + (m + n + 2)x + (3m + 3n + mn + 1)x 2 + (3m + 3n + 4mn)x 3 + (m + n + 6mn)x 4 + 4mnx 5 + mnx 6 + x m+n+2 Proof : Using the definition of clique polynomial, we obtain the following clique polynomials ω(t n, x) = 1 + nx + (n 1)x 2 ω(c n, x) = 1 + nx + nx 2 ω(f n, x) = 1 + (n + 1)x + (2n 1)x 2 + (n 1)x 3 ω(w n, x) = 1 + (n + 1)x + 2nx 2 + nx 3 Now, the result immediately follows from Theorem 35 Remark 38 In view of Corollary 37, some nonisomorphic graphs of the same order have the same convex subgraph polynomial Example 39 Consider for example the graphs P 3 +P 4 and P 3 +K 1,3 which are nonisomorphic graphs By Corollary 37, C(P 3 + P 4, x) = 1 + 7x + 17x x 3 + 6x 4 + x 7 = C(P 3 + K 1,3, x) 4 Composition of Graphs Definition 41 The composition G[H] of graph G with H is the graph with V (G[H]) = V (G) V (H) and (u, u )(v, v ) E(G[H]) if and only if either uv E(G) or u = v and u v E(H) Let us first illustrate the definition of composition of two graphs Consider the graphs P 2, P 3 and the compositions P 2 [P 3 ] and P 3 [P 2 ]

10 524 Ladznar S Laja and Rosalio G Artes, Jr Illustration 42 P 3 The compositions P 2 [P 3 ] and P 3 [P 2 ] of the graphs P 2 and P 2 : P 3 : a b P 2 [P 3 ] : (a, x) (a, y) (a, z) (b, x) (b, y) (b, z) (x, a) x y z (x, b) P 3 [P 2 ] : (y, a) (y, b) (z, a) (z, b) The next result determines the coefficient of clique polynomial of G [H] Lemma 43 Let G and H be connected graphs, and let 1 k ω(g)ω(h) ( ) ω(g) m Then ω k (G [H]) = ω m (G) ω jn (H) m=1 j 1 +j 2 +j m=k n=1 Proof : Let 1 k ω(g)ω(h), and let K be a complete subgraph of G [H] of order k By Theorem 25, V (K) = u A ({u} T u ), where A is complete subgraph of G and T u is complete subgraph of H for every u A Clearly, k = T u u A Suppose that A = 1 and A = {u} Then k = T u and there are ω 1 (G)ω k (H) of such K with A = 1 Suppose that A = 2 and A = {u, v} Then k = T u + T v and there exist positive integers j 1, j 2 ω(h) such that T u = j 1 and T v = j 2 so that k = j 1 + j 2 There are ω 2 (G) ω j1 (H)ω j2 (H) distinct complete subgraphs j 1 +j 2 =k graphs K relative to A with A = 2 In general, for A = m, there are ω m (G) j 1 +j 2 ++j m=k ( m ) ω jn (H) complete subgraphs K of G relative to A with A = m Therefore, ( ) ω(g) m ω k (G [H]) = ω m (G) ω jn (H) m=1 j 1 +j 2 +j m=k n=1 The next result uses Theorem 22[12] which is the characterization of convex sets in G [H] n=1

11 Convex subgraph polynomials of the join and the composition of graphs 525 Theorem 44 Let G and H be noncomplete connected graphs of orders p and q, respectively Then C (G [H], x) = ω(g, ω(h, x) 1) + x pq Proof : In view of Theorem 22, we have Thus, C (G [H], x) = 1 + ω (G [H], x) + x pq = 1 + It remains to show that 1 + Now, by Lemma 43, 1 + ω(g)ω(h) k=1 ω(g)ω(h) k=1 ω k (G [H]) x k = 1 + = 1 + ω 1 (G) + ω 3 (G) ω(g)ω(h) k=1 ω k (G [H)] x k = ω (G, ω(h, x) 1) ω(h) k=j 1 =1 ω(g)ω(h) k=1 ω(g) m=1 [ ω m (G) ω j1 (H)x j 1 + ω 2 (G) 3ω(H) + + ω m (G) k=j 1 +j 2 +j 3 =3 n=1 + + ω ω(g) (G) ω k (G [H]) x k + x pq j 1 ++j m=k n=1 2ω(H) k=j 1 +j 2 =2 n=1 3 ω jn (H)x j 1+j 2 +j 3 mω(h) k=j 1 +j 2 + +j m=m n=1 ω(g)ω(h) k=j 1 + +j ω(g) =ω(g) n=1 ] m ω jn (H)x k 2 ω jn (H)x j 1+j 2 m ω jn (H)x j 1+j 2 ++j m ω(g) ω jn (H)x j 1++j ω(g) ω(h) ω(h) = 1 + ω 1 (G) ω j (H)x j + ω 2 (G) ω j (H)x j j=1 ω(h) + ω 3 (G) ω j (H)x j j=1 ω(h) + + ω ω(g) (G) ω j (H)x j j=1 3 j=1 ω(h) + + ω m ω j (H)x j ω(g) = ω (G, ω(h, x) 1) j=1 2 m

12 526 Ladznar S Laja and Rosalio G Artes, Jr Corollary 45 (i) For m, n 3, C (T m [T n ], x) = 1 + mnx + [ m(n 1) + (m 1)n 2] x 2 + 2(m 1)n(n 1)x 3 + (m 1)(n 1) 2 x 4 + x mn (ii) For m 3 and n 4, C (T m [C n ], x) = 1 + mnx + [ mn + (m 1)n 2] x 2 + 2(m 1)n 2 x 3 + (m 1)n 2 x 4 + x mn (iii) For m 3 and n 4, C (C n [T m ], x) = 1 + nmx + [ n(m 1) + nm 2] x 2 + 2nm(m 1)x 3 + n(m 1) 2 x 4 + x nm (iv) For m, n 4, C (C m [C n ], x) = 1 + mnx + [ mn + mn 2] x 2 + 2mn 2 x 3 + mn 2 x 4 + x mn (v) For m, n 3, C (T m [F n ], x) = 1 + m(n + 1)x + [ (m 1)n 2 + 2mn + m 2 ] x 2 + [m(n 1) + 2(m 1)(n + 1)(2n 1)] x 3 + (m 1)(6n 2 4n 1)x 4 + 2(m 1)(n 1)(2n 1)x 5 + x m(n+1) (vi) For m 3 and n 4, C (T m [W n ], x) = 1 + m(n + 1)x + [ (m 1)(n + 1) 2 + 2mn ] x 2 + n(m + 4n + 4)x 3 + 2(m 1)n(3n + 1)x 4 + 4(m 1)n 2 x 5 + (m 1)n 2 x 6 + x m(n+1) (vii) For m 4 and n 3, C (C m [F n ], x) = 1 + m(n + 1)x + m(n 2 + 4n)x 2 + m(4n 2 + 5n 3)x 3 + m(2n 2 + 2n 3)x 4 + 2m(n 1)(2n 1)x 5 + m(n 1) 2 x 6 + x m(n+1) (viii) For m, n 4, C (C m [W n ], x) = 1 + m(n + 1)x + m(3n + 1)x 2 + mn(4n + 5)x 3 + 2mn(3n + 1)x 4 + 4mn 2 x 5 + mn 2 x 6 + x m(n+1) Proof : To prove (i), we note that ω(t m, x) = 1 + mx + (m 1)x 2 and ω(t n, x) = 1 + nx + (n 1)x 2

13 Convex subgraph polynomials of the join and the composition of graphs 527 Thus, applying Theorem 44, we get C (T m [T n ], x) = 1 + m(nx + (n 1)x 2 ) + (m 1)(nx + (n 1)x 2 ) 2 + x mn = 1 + mnx + [m(n 1) + (m 1)n 2 ]x 2 + 2(m 1)n(n 1)x 3 + (m 1)(n 1) 2 x 4 + x mn The proof of (ii)-(x) are similar to (i), using the following clique polynomials ω(t n, x) = 1 + nx + (n 1)x 2 ω(c n, x) = 1 + nx + nx 2 ω(f n, x) = 1 + (n + 1)x + (2n 1)x 2 + (n 1)x 3 ω(w n, x) = 1 + (n + 1)x + 2nx 2 + nx 3 The above result provides another type of nonisomorphic graphs of the same order having the same convex subgraph polynomial representation Example 46 Consider the path P 4 and the star graph K 1,3 Since ω(p 4, x) = ω(k 1,3, x), it follows that ω (P 4 [C 4 ], x) = ω (K 1,3 [C 4 ], x), so that C (P 4 [C 4 ], x) = C (K 1,3 [C 4 ], x) But the two compositions P 4 [C 4 ] and K 1,3 [C 4 ] are non isomorphic References [1] Rosalio G Artes, Jr and Ladznar S Laja, Zeros of Convex Subgraph Polynomials, Applied Mathematical Sciences, 8 (2014), no 59, [2] Saieed Akbari and Mohammad Reza Oboudi, On the Edge Cover Polynomial of a Graph, European Journal of Combinatorics, 34 (2013), no 2, [3] A Ali Ali and Walid A M Said, Wiener Polynomials for Steiner Distance of Graphs, J J Appl Sci, 8 (2006), no 2, [4] Saeid Alikhani and Torabi Hamzeh, On the Domination Polynomials of Complete Partite Graphs, World Applied Sciences Journal, 9 (2010), no 1, [5] Saeid Alikhani and M H Reyhani, On the Values of Independence and Domination Polynomials at Specific Points, Transactions on Combinatorics, 1 (2012), no 2, [6] S Beraha, J Kahane and N J Weiss, Limits of Chromatic Zeros of Some Families of Maps, Journal of Combinatorial Theory, Series B, 28 (1980),

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