Degree-based indices of hypergraphs: Definitions and first results

For ak-uniform hypergraphH, we introduce degree-based indices such as the generalsum-connectivity index chi(a)(H) =& sum;(v1v2)<middle dot><middle dot><middle dot>(vk is an element of E(H))[d(v(1)) +d(v(2)) +<middle dot><middle dot><middle dot>+d(v(k))](a) and thegeneral Randi c index Ra(H) =& sum;(v1v2)<middle dot><middle dot><middle dot>v(k)is an element of E(H)[d(v(1))d(v(2))<middle dot><middle dot><middle dot>d(v(k))](a), where a is an element of R,E(H) is the set of hyperedges of Hand d(v(i)) is the degree of a vertex v(i) in H;k >= 2 and i= 1,2,...,k. Other indices such as the first and second Zagreb index, first andsecond hyper-Zagreb index, classical sum-connectivity index, classical Randi c index andharmonic index of a hypergraphHare special cases of the general indices. Fora >0,we obtain upper bounds on chi(a)(H) and R-a(H) for a uniform hypergraphHwith givenorder, order and number of isolated vertices, order and maximum degree, order anddiameter at least 2, and lower bounds for uniform hypergraphs with given order and noisolated vertices, order and minimum degree, and order and maximum possible degree.We also present extremal graphs for all the bounds. Bounds on Zagreb indices followfrom our results on general indices.