Research On The Problems Of2-factor, Supereulerianity And Topological Index In Graphs

In this dissertation, we systematically study some problems about the supereuleri-anity,2-factor and several topological indices of graphs, including the supereulerianityof graphs, the existence and the number of components of2-factors in claw-free graphs,and the extremal problems related to the degree and distance in graphs. The wholethesis is divided into eight chapters.In Chapter1, a concise survey to the background and history is presented forsupereulerianity,2-factor of graphs and topological indices. The structure, the problemsto be studied and the main results are also briefy summarized. In Section1.3, wepresent some terminologies and notations.In Chapter2, we study the supereulerianity of graphs. We frst give a characteri-zation of collapsible of a graph which is2-edge-connected simple graph and α (G)≤2,which strengthens the main result in [H.-J. Lai, H. Yan, Supereulerian graphs andmatchings [J]. Appl. Math. Lett.24(2011)1867-1869]. And we also prove a con-jecture in the same paper above, that if G is a3-edge-connected simple graph andα (G)≤5, then G is supereulerian if and only if G is not contractible to the Petersengraph.In Chapter3, we study the existence of2-factors in claw-free graphs. We frstpropose two operations on a graph preserving the (non)existence of2-factors in its linegraph. By using these two operations, we present a characterization for a graph G tohave a2-factor in its line graph L(G). Then by applying to the new characterization,we give a sufcient condition for a claw-free graph has a2-factor, which generalizes thetwo former results and is best possible.In Chapter4, we study the existence of2-factors in claw-free graphs with locallydisconnected vertices. We prove that if a connected claw-free graph G of order at leastthree satisfes the following two conditions:(1) Every locally disconnected vertex ofdegree at least3lies on an induced cycle of length at least4with at most s edges which lie on triangles and with at least s5locally connected vertices, for some nonnegativeinteger s;(2) Every locally disconnected vertex of degree2lies on an induced cycle Cwith at most s edges which lie on triangles and with at least s3locally connectedvertices such that G[V (C)∩V2(G)] is a path or a cycle, for some nonnegative integers, then G has a2-factor. Our results are best possible and extend a prior result.In Chapter5, we study the geometric-arithmetic index of a graph. By defninga new class of geometric-arithmetic index which is named the k-ordinary geometric-arithmetic index, we mainly consider the basic mathematical properties of this newindex and discuss the diference and relationship between this index with the knowngeometric-arithmetic index GA in physicochemical properties.In Chapter6, we study the general sum-connectivity index of polyomino chains.After giving an efcient formula for computing the general sum-connectivity index ofpolyomino chains, we characterize the extremal polyomino chains with respect to thisindex, which generalizes one of the main results in [Z. Yarahmadi, A. Ashraf, S. Moradi,Extremal polyomino chains with respect to Zagreb indices [J]. Appl. Math. Lett.25(2012)166-171].In Chapter7, we study the degree distances of four sums of two graphs, andestablish two upper bounds for the degree distances of four sums of two graphs in termsof other indices of two individual graphs.In the last chapter, we present some open problems related to our dissertation.