# Linear Algebra and its Applications

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2 1394 F. Liu et al. / Linear Algebra and its Applications ) e.g., [3, p. 157]) and distance regular graphs have attracted more attention, see [1,] forexamples. It is also proved that the Cartesian product of an odd cycle with the complete graph K 2 is DAS [12]. In this note, we investigate the structure of cubic graphs of given order that have maximum number of triangles, and characterize the extremal cubic graphs. Four families of cubic graphs with maximum number of triangles are determined to be DAS. Recall that K n and C n denote the complete graph and cycle on n vertices, respectively. Let K 4 denote the graph obtained from K 4 by deleting an edge. Denote by G a G b ) the graph obtained from C 3 C 4 ) by appending a pendant edge, G c the bowtie graph two triangles share a common vertex). The theta graph θa, b, c) is the graph consisting of three internally disjoint paths with common endpoints and lengths a + 1, b + 1, c + 1 and with a b c 0. The dumbbell graph Dd, e, f ) which is formed by joining two disconnected cycles C d+1 and C e+1 by a path P f +2 two endpoints of P f +2 are coalescing to one vertex of C d+1 and one vertex of C e+1, respectively.) where d e 1, f 0. For a vertex v VG), Nv) is the neighbors of v, i.e., Nv) ={u VG) uv EG)}, and N[v] =Nv) {v}. Let x k be the number of vertices of degree k of a graph G, then we write the degree sequence of G as d G = 0 x 0, 1 x 1,...,k x k,..., x ). The notion and symbols not defined here are standard, see [3]for any undefined terms. 2. Cubic graphs with maximum number of triangles In this section we characterize cubic graphs with maximum number of triangles. Moreover, we determine four families DAS extremal graphs. Let G G K 4 ) be a connected cubic graph with vertex set VG) and edge set EG). Clearly, each vertex of a cubic graph G belongs to at most two triangles. We call a vertex v i is an i-type vertex of G if v i is exactly contained in i triangles i = 0, 1, 2). DenotebyV i G) the set of i-type vertices in G and n i = V i G), then the vertex set VG) can be partitioned into VG) = V 0 G) V 1 G) V 2 G). By definition, for v V 2 G), N[v] induces K 4 in G in which v and v belong to exactly two triangles and u and u belong respectively to one triangles see Fig. 1), such an induced subgraph is said to be a K 4 -part of G. Thuswehavetheclaimbellow. Claim 1. Let G G K 4 ) be a connected cubic graph, then n 2 is even, G contains exactly n 2 2 numbers of K 4 -part and n 1 n 2. If x V 1 G) then N[x] ={x, y, z, w} induces a G a where x, y and z are all contained in a common triangle see Fig. 1), which we call a C 3 -part of G, and w lies in at most on triangle. Thus we have the claim bellow. Claim 2. n 1 n 2 0 mod 3). Since a cubic graph G except for K 4 )withn vertices consists of some K 4 -parts, C 3-parts and some vertices belonging to V 0 G). IftheK 4 parts and C 3 parts are respectively contracted as a vertex then we get a cycle or a 2, 3)-bidegree graph G from G. For example, in Fig. 1 a)-g) we illustrate all cubic graphs consisting of four numbers of K 4 -parts and two vertices contained in no triangles, and a )-e ) the cubic graphs consisting of three numbers of K 4 -parts and two numbers of C 3-parts. In general we denote by G n k,l,n 4k 3l)) the cubic graphs with n vertices consisting of k numbers of K 4 -parts, l numbers of C 3 -parts and n 4k 3l) vertices lying in no triangles. For instance, the graphs shown in Fig. 1 a)-g) are the graphs in G 18 4, 0, 2), and a )-e ) the graphs in G 18 3, 2, 0), where the small thick dots are the vertices belonging to V 0 G), big thick dots denote the subgraphs K 4 and big hollow dots denote the subgraphs K 3, such graphs shown in Fig. 1 are called the part-contracted graphs of cubic graphs. Note that these graphs may have cycles of length two). It is worth mentioning that G n k, 0, 0) contains exactly one part-contracted graph i.e., a cycle C k ), and the part-contracted graph of a cubic graph will be itself if it contains no triangles. It is easy to verify the following two claims. Claim 3. If G G n k,l,n 4k 3l) then G contains exactly 2k + l triangles.

3 F. Liu et al. / Linear Algebra and its Applications ) Fig. 1. G 18 4, 0, 2) and G 18 3, 2, 0). Claim 4. Let G G K 4 ) be a connected cubic graph, then the number of triangles of G is tg) = i=0 in i. The following theorem characterizes all connected cubic graphs with maximum number of triangles. Theorem 2.1. Let G be a connected cubic graphs with n = 2r vertices, m = 3r edges r 3) and maximum number of triangles. If r is even then G = G r, 0, ) n 2 0 ;ifrisoddtheng G r 1, 0, ) n 2 2 or G r 3, 2, ) n 2 0. Proof. To determine such an extremal cubic graph G with maximum number of triangles is equivalent to solving the following optimization problem: max tg) = 1 3 n 1 + 2n 2 ), { n0 + n 1 + n 2 = VG) =2r s.t. n i = V i G) 0 i = 0, 1, 2.) 1) From 1) we see that tg) is a decreasing function in n 0 and an increasing function in n 2 n 1.A natural idea is when tg) achieves its maximum then n 0 is smallest and n 2 is largest or nearly largest by nearly largest we mean the value is one less than the maximum). Hence we need to consider the largest n 2. Suppose that G is an extremal cubic graph satisfying 1). From Claim 1 we get the following inequalities: n 2 0 mod 2) n 1 n 2 2) n 0 0 Combing 1) and 2), we obtain 2r n 0 n 1 = n 2 n 1 n 1 + n 0 = 2r n 2,thus n 2 r. 3) If r is even then n 2 = r achieves maximum value. By Claim 1, r = n 2 n 1 n n 2 = r, which implies n 0 = 0, and so G is a graph in G n r, 0, 0). HencetG) = r achieves its maximum value.

4 139 F. Liu et al. / Linear Algebra and its Applications ) If r is odd then n 2 = r 1 achieves maximum value. By Claim 2, n 1 = r 1 and so n 0 = 2, we obtain a solution tg) = r 1for1) and find G G r 1, 0, ) n 2 2 see Fig. 1 for example). Next, if n 2 = r 3 takes nearly largest value then G contains r 3 numbers of K 2 4 -parts. By Claim 2 and Claim 3 we know G G r 3, 2, ) 2 0 has maximum number of triangles: tg) = r = r 1. Finally, if n 2 = r 2s + 1)s 2) then G has exactly r 2s+1) numbers of K 2 4 -parts. By Claim 2 and Claim 3, G has at most r 2s + 1) + 22s+1) triangles, but r 2s + 1) + 22s+1) < r 1, and so tg) 3 3 cannot achieves its maximum value if n 2 = r 2s + 1)s 2). Theorem 2.2. The order of the set G n k, 0, 2) and G n k 1, 2, 0) k k2k+3) k 1)2k+1) 2)are and + 1, respectively. Proof. Denote by p i k) the number of the unordered partitions of k into i parts. We have p 1 k) = 1 and p 2 k) = k. It is known in [13, p. 152] that p 2 3 k) = k 2 +.LetG be a graph in G 12 n k, 0, 2) and G be its part contracted graph. Since d G = 3 2, 2 k ), G S1 ={θa, b, c) a + b + c = k} or S 2 ={Dd, e, f ) d + e + f = k}. FortheorderofS 1,weobtainifc = 0, then S 1 =p 2 k); ifc > 0, then S 1 =p 3 k). For the cardinality of S 2, we conclude if f = 0, then S 2 =p 2 k); iff > 0, then S 2 = k 2 j=1 p 2k j). Hence direct calculation shows that k 2 k2k G n k, 0, 2) = S 1 + S 2 =p 2 k)+p 3 k))+ p2 k) + p 2 k j) + 3) =. 4) Let H be the part contracted graph of the graph H in Gn k 1, 2, 0).If H {θa, b, c) a + b + c = k 1}, unlike the graph in G n k, 0, 2), the unique different case a = k 1, b = c = 0 may occur. For the other cases, the argument is exactly the same as the above, it suffices to substitute k 1into4) k 1)2k+1) and which gives.henceweget G n k 1, 2, 0) = k 1)2k+1) + 1. Denote by 1 a, b, c), 2 a, b, c ) the cubic graphs in G n k, 0, 2) and G n k 1, 2, 0) whose part contracted graphs are θ 1 a, b, c) and θ 2 a, b, c )a + b + c = k, a + b + c = k 1), respectively. Denote by D 1 d, e, f ), D 2 d, e, f ) the cubic graphs in G n k, 0, 2) and G n k 1, 2, 0) whose part contracted graphs are Dd, e, f ) and Dd, e, f )d+e+f = k, d +e +f = k 1), respectively. Now we turn to the spectral characterization of graphs G n k, 0, 0)k 2) and D 1 1, 1, k 2), 1 k 1, 1, 0), 2 k 1, 0, 0). Let G G n k, 0, 2) G n k 1, 2, 0) and H be cospectral with G. WeobtainH is also a connected cubic graph with 2k numbers of triangles [4]. Hence H G n k, 0, 2) G n k 1, 2, 0).IfG = G n k, 0, 0), Theorem 2.1 implies G n k, 0, 0) is the unique cubic graph with 4k vertices and 2k triangles, it follows that G n k, 0, 0) is DAS. Theorem 2.3. G n k, 0, 0)k 2) is determined by its adjacency spectrum. In order to show that the graphs D 1 1, 1, k 2), 1 k 1, 1, 0), 2 k 1, 0, 0) are also DAS, we enumerate the number of closed walks of length 4, 5, ofgraphsing n k, 0, 2) and G n k 1, 2, 0) to rule out their possible cospectral mate. Let N G H) be the number of subgraphs of a graph G which are isomorphic to H and let N G i) be the number of closed walks of length i in G. The following lemma provides some formulae for calculating the number of closed walks of length 4, 5, foranygraph. Lemma 2.4 [10,11]. The number of closed walks of length 4, 5, of a graph G are giving in the following: i) N G 4) = 2 EG) +4N G P 3 ) + 8N G C 4 ). ii) N G 5) = 30N G K 3 ) + 10N G C 5 ) + 10N G G a ), j=1

5 F. Liu et al. / Linear Algebra and its Applications ) iii) N G ) = 2 EG) +12N G P 3 ) + N G P 4 ) + 12N G K 1,3 ) + 24N G K 3 ) + 48N G C 4 ) + 3N G K 4 ) + 12N G G b ) + 24N G G c ) + 12N G C ), Theorem 2.5. D 1 1, 1, k 2), 1 k 1, 1, 0) and 2 k 1, 0, 0)d 1, k 2) are determined by their adjacency spectrum. Proof. It suffices to show D 1 1, 1, k 2), 1 k 1, 1, 0) and 2 k 1, 0, 0) have no cospectral mate in G n k, 0, 2) G n k 1, 2, 0). We consider the number of closed walks of length 4 of a graph G in G n k, 0, 2) G n k 1, 2, 0). Since EG) =32k+1) and N G P 3 ) = ) dv) v VG) 2 = 2k+1), by Lemma 2.4 i), NG 4) = 0k+ 30+8N G C 4 ).ThusN G 4) is determined by N G C 4 ). From the graph structure we see that N D1 1,1,k 2) C 4 ) = k + 4, N D1 d,1,k d 1)C 4 ) = k + 22 d k 1), N 1 a,b,c)c 4 ) = N D1 d,e,f )C 4 ) = k 2 e d), N 2 k 1,0,0)C 4 ) = k. ForanygraphG G n k 1, 2, 0) \{ 2 k 1, 0, 0)}, N G C 4 ) = k 1. Thus D 1 1, 1, k 2) is DAS. For 1 k 1, 1, 0) and 2 k 1, 0, 0), we show that they have no cospectral mate in the set S 3 ={ 1 a, b, c), 2 k 1, 0, 0), D 1 d, e, f )2 e d)}. We need to count the number of closed walks of length 5 and. For a graph G in S 3,SinceN G K 3 ) = 2k and N G G a ) = k, by Lemma 2.4 ii) N G 5) = 120k + 10N G C 5 ) is determined by N G C 5 ).Clearly,N 1 k 1,1,0)C 5 ) = N 2 k 1,0,0)C 5 ) = 2. For G S 3 \{ 1 k 1, 1, 0), 2 k 1, 0, 0)}, it is easy to verify that N G C 5 ) = 0. Thus 1 k 1, 1, 0) and 2 k 1, 0, 0) are not cospectral with the graphs in S 3 \{ 1 k 1, 1, 0), 2 k 1, 0, 0)}. Finally we enumerate the number of closed walks of length of graphs 1 k 1, 1, 0) and 2 k 1, 0, 0). Since 1 k 1, 1, 0) and 2 k 1, 0, 0) have the same number of edges, P 3, K 3, K 1,3, P 4 N 1 k 1,1,0)P 4 ) = 4 E 1 k 1, 1, 0)) 3N 1 k 1,1,0)K 3 ) = 4 E 2 k 1, 0, 0)) 3N 2 k 1,0,0) K 3 ) = N 2 k 1,0,0)P 4 )), C 4 and G c,moreovern 1 k 1,1,0)K 4 ) = k, N 2 k 1,0,0)K 4 ) = k 1; N 1 k 1,1,0)G b ) = 2k, N 2 k 1,0,0)G b ) = 2k + 2; N 1 k 1,1,0)C ) = 2, N 2 k 1,0,0)C ) = 1. By Lemma 2.4 iii), N 1 k 1,1,0)) N 2 k 1,0,0)) = 24. Therefore 1 k 1, 1, 0) and 2 k 1, 0, 0) are not cospectral, and they are determined by their adjacency spectrum. Acknowledgment The authors would like to thank the anonymous referees for their valuable suggestions used in improving this paper. References [1] S. Bang, E.R. van Dam, J.H. Koolen, Spectral characterization of the Hamming graphs, Linear Algebra Appl ) [2] B. Blulet, B. Jouve, The lollipop graph is determined by its spectrum, Electron. J. Combin ), Research Paper 74. [3] D.M. Cvetković, M. Doob, H. Sachs, Spectra of Graphs, Second ed., VEB Deutscher Verlag der Wissenschaften, Berlin, [4] E.R. van Dam, W.H. Haemers, Which graphs are detemined by their spectrum?, Linear Algebra Appl ) [5] E.R. van Dam, W.H. Haemers, Developments on the spectral characterizations of graphs, Discr. Math ) [] E.R. van Dam, W.H. Haemers, J.H. Koolen, E. Spence, Characterizing distance-regularity of graphs by the spectrum, J. Combin. Theory Ser. A ) [7] M. Doob, W.H. Haemers, The complement of the path is determined by its spectrum, Linear Algebra Appl ) [8] N. Ghareghani, G.R. Omidi, B. Tayfeh-Rezaie, Spectral characterization of graphs with index at most 2 + 5, Linear Algebra Appl ) [9] W.H. Haemers, X. Liu, Y. Zhang, Spectral characterizations of lollipop graphs, Linear Algebra Appl ) [10] F.J. Liu, Q.X. Huang, J.F. Wang, Q.H. Liu, The spectral characterization of -graphs, Linear Algebra Appl ) [11] G.R. Omidi, On a signless Laplacian spectral characterization of T-shape trees, Linear Algebra Appl ) [12] F. Ramezani, B. Tayfeh-Rezaie, Spectral characterization of some cubic graphs, Graphs and Combinatorics, DOI / s [13] J.H. van Lint, R.M. Wilson, A Course in Combinatorics, Cambridge University Press, [14] W. Wang, C.X. Xu, On the spectral characterization of T-shape trees, Linear Algebra Appl )

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