(10)The equality holds if and only if G is regular graph. ��(G)��max?uv��E(G)du(du+mu)+dv(dv+mv)2.(11)The equality holds if and only if G is regular graph. ��(G)��max?u��V(G)du+dumu2.(12)The equality holds if and only if G is regular graph. ��(G)��max?uv��E(G)du+dv+(du?dv)2+4mumv4.(13)The equality holds if and only if G is regular graph.2.2. Main ResultsAll of these upper bounds mentioned in Pazopanib Section 2.1 are characterized by the degree and the average 2-degree of the vertices. Actually, we can also use other invariants of the graph to estimate the spectral radius. In the following, such an invariant will be introduced. In a graph, a circle with length 3 is called a triangle. If u is a triangle’s vertex in a graph, then u is incident with this triangle. Denote by Tu the number of the triangles associated with the vertex u.
For example, in Figure 1, we have Tu = 3 and Tv = Tw= 0.Figure 1Graph with triangles.Let Nu��Nv be the set of the common adjacent points of vertex u and v; then |Nu��Nv| present the cardinality of Nu��Nv.Now, some new and sharp upper and lower bounds for the spectral radius will be given.Theorem 5 ��Let G be a simple connected graph with n vertices. Then��(G)��max?uv��E(G)du2mu+dv2mv?2(Tu+Tv)2(dudv?|Nu��Nv|);(14)the equality holds if and only if G is a regular graph.Proof ��Let K = diag (dudv ? |Nu��Nv | :uv E(G)) is a diagonal matrix and B is the adjacency matrix of the line graph. Denote N = K?1BK, then N and B have the same eigenvalues. Since G is a simple connected graph, it is easy to obtain that N is nonnegative and irreducible matrix.
The (uv, pq)th entry of N is equal to{dpdq?|Np��Nq|dudv?|Nu��Nv|,pq~uv,0,else,(15)here pq ~ uv implies that pq and uv are adjacent in graph. Hence, the uvth row sum Ruv(N) of N =du2mu+dv2mv?2(Tu+Tv)dudv?|Nu��Nv|?2.(16)From?=du2mu+dv2mv?2dudv?2(Tu+Tv)+2|Nu��Nv|dudv?|Nu��Nv|??��q~u|Nu��Nq|??+��p~v|Np��Nv|?2|Nu��Nv|dudv?|Nu��Nv|??=��q~ududq+��p~vdpdv?2dudvdudv?|Nu��Nv|?is��pq~uvdpdq?|Np��Nq|dudv?|Nu��Nv| Lemmas 1 and 2, we have��(G)��12��(B)+1��max?12Ruv(N)+1:uv��V(H).(17)It means that (14) holds and the equality in (14) holds if and only if G is a regular graph. In a graph, let �� and �� represent the number of vertices with the maximum degree and minimum degree, respectively. Then, we get the following results.Theorem 6 ��Let G be a simple connected graph with n vertices. If �� �� min n ? 1 ? ��, n ? 1 ? ��, then��(G)��2m+��(�ġ�?1)?�¦�?(n?1?��)�ġ�,(18)��(G)��2m+(����?1)��?����?(n?1?��)����;(19)the Dacomitinib equality holds if and only if G is a regular graph.Proof ��Since Rv(A2) is exactly the number of walks of length 2 in G with a starting point v, thusRv(A2)=��u~vdu=2m?dv?��u?vdu.(20)Therefore, from Lemmas 1 and 3, if �� �� n ? 1 ? ��, we have dv �� n ? 1 ? �� for any v V(G).