 to present a talk here. So, the title of my talk is symbolic powers of age ideals of weighted oriented graphs. You have already seen our talk in the same topic given by Manohar in the morning session. So, I will continue. He has already set up many things for me. So, I will first start with the definition of the symbolic power. So, in an Oetherian ring if i is an ideal then the nth symbolic power is defined as in this way where we are considering the extension of this in the localization and then contracting it back and we are varying this intersection over all the associated primes. So, this is the more general definition of the symbolic power and then there are certain properties that symbolic power satisfies. First of all that the ordinary power is always sitting inside the symbolic power and then if a is written equal to b in that case nth symbolic power is sitting inside the bth symbolic power and we have this property as well. Now, in general this ordinary power is sitting inside the symbolic power, but it may not be an equality. So, why the symbolic power is important that is because of this theorem by Nagata and Zyruski it gives you some geometrical importance why the symbolic powers are important. So, Nagata and Zyruski prove that if you are in a if you have a perfect field and i is an radical ideal in this polynomial ring in that case the nth symbolic power is nothing but the collection of all monomials all polynomials which vanish up to order in along this variety of this ideal. So, that is why symbolic powers catches more information than the ordinary powers and that is why it is important to study. So, now as the title suggests we are studying the symbolic power of weighted oriented graph. So, let me give you first some graph theoretic preliminaries for this. So, first I will talk about simple graph and then I will go into the weighted oriented graph. So, what is a simple graph? A simple graph is nothing but a ordered pair where we are having this two sets one is the set of vertices that I have denoted by v of g and e g denotes the set of edges which is nothing but the two elements subsets of this set of vertices. Now when and so in this talk what we are trying to do we have this we are trying to consider the algebraic objects coming from the graph theory. So, we start with a graph and then with that graph we will try to associate one ideal and then we will study the invariance of algebraic invariance of that ideal using the graph theoretic invariance of that graph. So, the ideal that we will study here is the age ideal. So, what is the age ideal? So, age ideal is generated if you start with a graph g then you consider first the polynomial ring in as many variables as many vertices you are having in the graph and then in that polynomial ring you define this ideal i g which is generated by the edges of this graph. So, this algebraic object we will study the properties of this ideal using the properties of the graph theoretical invariance. So, let us see one example here first. So, here this graph is this triangle x 1, x 2, x 3 it has three vertices. So, then corresponding age ideal will be generated by this three edges x 1, x 2, x 2, x 3 and x 3, x 1. So, this was the simple graph and it has the age ideal of a simple graph has already been studied a lot in the literature and many things are already known. In fact, the description of the symbolic powers are also known for some classes and regularity and all those things has been studied extensively. So, recently we are concentrating on this weighted oriented graph. So, what is weighted oriented graph? So, weighted oriented graph is a triplet where we have the set of vertices and also we have edges and then we have one function which is weight function. So, we are giving weight now to each of the vertices and not only that we have we are giving also an orientation to each of the edges. So, that is how we are considering this weighted oriented graph and then we are associating age ideal corresponding to this weighted oriented graph which is defined as x i, x j, w j, where w j is the weight of the vertex x j and this we have an orientation to this edge. So, the edge is directed from x i to x j, then we will consider the generator to be x i, x j to the power w j. So, this is the age ideal we are considering for this graph. So, let us see one example here. So, both of this graphs are basically triangle, but now in the first graph D1 you can see that I have given weights 2 to each of this vertices and I have this arrows in this particular direction. So, for the first one you can see the age ideal is generated by x 1, x 2 square as we are having an arrow from x 1 to x 2 and then we are having arrow from x 2 to x 3. So, that is why we are having the generator x 2, x 3 square and the other one we are having the arrow from x 3 to x 1. So, that is what we are getting this generator x 3, x 1 square. Whereas, in the second graph we have given this x 1, the weights of x 1 we have given 1 and in that case you can see this will be the age ideal and in this case you can see that if we change the direction, if we change the direction of the arrows or the weights of the vertices, then the age ideal changes. So, it is very much dependent on, so it is very much dependent on the arrows as well as on the weights. So, it is that is why it is difficult to study as well. Now, we will try to give simpler description of this symbolic powers. So, the description of the symbolic powers for ordinary graph was already known. So, the following theorem by Boschian in 2016, he proved that if you are in a graph and i is the age ideal and this v 1, v 2, v r are the minimal vertex cover, then this i can be written as intersection of this prime ideals where this prime ideals corresponds to this minimal vertex cover. So, this with this decomposition we see that the emits symbolic power of i has this simpler expression which is nothing but the power of the primes, emits power of the primes and their intersection. So, whatever the definition we have seen in the beginning for symbolic powers, now it simplifies if we are considering the age ideal. So, now we have seen that for ordinary graph this minimal vertex cover plays an important role in order to describe the symbolic power. So, we try to investigate what will play a role if I consider the age ideal of a weighted oriented graph. And in that case we have seen that this minimal vertex cover we have to replace with strong vertex cover of the graph and in that case. So, if we replace this minimal vertex cover with strong vertex cover then we have some kind of similar result that we were having for the simple graph. And then we Cooper in 2017 have proved that in this case also we have a simpler description of the symbolic power. So, this SS symbolic power can be written as intersection of this powers of this ideals which are coming from the basically the corresponding to the maximal strong vertex covers of this graph T. Now, the question that we are interested in this when the symbolic power and ordinary power are equal for all values of S. And for simple graph this was a famous result by Aaron Simis. He proved that the ordinary and symbolic power are equal for all is written equal to 1 if and only if G is bipartite. So, it completely classifies the class of graph for which this will be equal and this is necessary and sufficient condition also. So, then we try to investigate whether we can give some similar criteria for the age ideal of the weighted oriented graph or not. And in this direction we have first proved that if D is a weighted oriented graph such that the vertex set is a strong vertex cover in that case the symbolic power and ordinary powers are equal. But then we ask whether the converse is also true or not. And then we see that under certain condition it will be that I will come to that later. So, it is already known that if you have odd cycle present in a graph then of length 2n plus 1 in that case n plus 1 with ordinary and symbolic power are different. We see that this if this will be true in the for the age ideal of the weighted oriented graph under certain condition. So, if the vertex set of this cycle of length 2n plus 1 which is present in this weighted oriented graph satisfies this condition in that case also the similar result like the for the underlined graph will holds true for the weighted oriented graph as well. And we prove that if you your graph as every age of this weighted oriented graph lies in some induced odd cycle in that case the whole vertex set is a strong vertex cover if and only if the symbolic powers and ordinary powers are equal. We have already proved that if Vd is a strong vertex cover in this case equality holds. Now we are saying that the opposite the other direction is also true but with the condition that each age of d lies in some induced odd cycle. And as an application of this theorem we are able to see that in odd cycles complete graphs click some of finite number of odd cycles and complete graphs and complete imparted graphs in all these classes of graphs each age lies in some induced odd cycles. So, we can apply this theorem for all this class of graphs. Next we have proved that if we are having an unicyclic graph with a unique odd cycle of length 2n plus 1 in that case the symbolic powers and ordinary powers will be equal for all s greater than equal to 2 if and only if the weight of those vertices which whose degree is greater than equal to 2 has to be greater than equal to 2. In this case also this equality holds and it is an if and only if or as well. Then we also have in the further direction we have proved that previous result was for unicyclic graph and this one we have seen that if you have a naturally oriented even cycle in that case of length not equal to 4. In that case this equality holds if and only if all the vertices of D have non-trivial weights that means all the vertices of this graph has to have weight greater than equal to 2. And for length of cycle 4 in this case this will be equality if one of these conditions are satisfied either all the vertices have non-trivial weights or one vertex has non-trivial weight or only two non-consecutive vertices have non-trivial weights. Further in order to compare this symbolic power of ordinary graph and the weighted oriented graph we have used this map. So, we have defined this map from the polynomial ring to itself where V plus D is the state of vertices which are sinc. The sinc has already been defined in the talk of Manohar and U is a subset of this V plus D. In that case we are defining this map we are sending Xj to Xj if Xj does not belongs to U and Xj to Xj to the power Wj if Xj belongs to U. So, in this way what we are trying to do is that if you see that if I take my U to be equal to V plus D in that case I is the edge ideal of this weighted oriented graph and D prime becomes the underline graph and I tilde becomes the edge ideal of the underline graph. And in that case we have seen that this ordinary powers and the symbolic powers behaves in a similar way. And using this map we are able to prove that the edge ideal of the weighted oriented for the weighted oriented graph symbolic power and the ordinary power will be equal if and only if they are equal for the underline graph. And similarly using this result we are able to prove the generalization of Simi's result for weighted oriented graph. We have proved that D with a weighted oriented graph for the vertices of V plus D are sinc then G is bipartite if and only if ordinary and symbolic powers are equal for all is greater than equal to 2. So, here we need the assumption that V plus D vertices of V plus D are sinc. Then similar result of Simi's holds here as well and we have also seen that if we are in a star graph in that case also the symbolic powers and ordinary powers are equal. So, this gives the equality of symbolic and ordinary powers for certain classes of graph which sometimes gives the necessary and sufficient conditions also. And still the description of the symbolic power in general it is not known for weighted oriented graphs. And there are many questions which are still open in this area and that is all I would like to say. So, these are the references.