Transactions on Combinatorics
Volume:5 Issue: 4, Dec 2016

The vertex PI index PI(G)=∑xy∈E(G)[nxy(x)ýýnxy(y)] is a distance-based molecular structure descriptorý, ýwhere nxy(x) denotes the number of vertices which are closer to the vertex x than to the vertex y
and which has been the considerable research in computational chemistry dating back to Harold Wiener in 1947ý. ýA connected graph is a cactus if any two of its cycles have at most one common vertexý. ýIn this paperý, ýwe completely determine the extremal graphs with the greatest and smallest vertex PI indices mong all cacti with a fixed number of verticesý. ýAs a consequenceý, ýwe obtain the sharp bounds with corresponding extremal cacti and extend a known resultý.

Let R be a commutative ring with unity. The comaximal ideal graph of R, denoted by C(R), is a graph whose vertices are the proper ideals of R which are not contained in the Jacobson radical of R, and two vertices I1 and I2 are adjacent if and only if I1=R. In this paper, we classify all comaximal ideal graphs with finite independence number and present a formula to calculate this number. Also, the domination number of C(R) for a ring R is determined. In the last section, we introduce all planar and toroidal comaximal ideal graphs. Moreover, the commutative rings with isomorphic comaximal ideal graphs are characterized. In particular we show that every finite comaximal ideal graph is isomorphic to some C(Zn).

In this paperý, ýwe initially introduce the concept of n-distance-balanced property which is considered as the generalized concept of distance-balanced propertyý. ýIn our considerationý, ýwe also define the new concept locally regularity in order to find a connection between n-distance-balanced graphs and their lexicographic productý. ýFurthermoreý, ýwe include a characteristic method which is practicable and can be used to classify all graphs with i-distance-balanced properties for i=2,3 which is also relevant to the concept of total distanceý. ýMoreoverý, ýwe conclude a connection between distance-balanced and 2-distance-balanced graphsý.

The first Zagreb indexý, ýM1(G)ý, ýand second Zagreb indexý, ýM2(G)ý, ýof the graph G is defined as M1(G)=∑v∈ýýV(G)d2(v) and M2(G)=∑e=uv∈E(G)d(u)d(v), whereý ýd(u) denotes the degree of vertex uý. ýIn this paperý, ýthe firstý ýand second maximum values of the first and second Zagreb indicesý ýin the class of all n−vertex tetracyclic graphs are presentedý.

In this paperý, ýby studying three q-Catalan identities given by Andrewsý, ýwe arrive at a certain number of congruencesý. ýThese congruences are all modulo Φn(q)ý, ýthe n-th cyclotomic polynomial or the related functions and modulo q-integersý.