Graphs of $P_3$ My definition of $P_3$ is a graph with $4$ vertices and $3$ edges arranged in a linear fashion. I know it is drawn like this on LHS, but is the RHS acceptable too? :


The most important thing to know about graphs is that the graph is not the diagram. The definition of a graph is "a pair consisting of a set $V$ of vertices and a set $E$ of edges where each edge is an unordered pair of vertices" or something similar - there are several subtly different variations for different settings, they're not important for now. So a path graph would formally be an object like $(\{a,b,c,d\}, \{ab, bc, cd\})$.
We can represent this graph by a diagram in any number of ways: all of the diagrams below are valid representations of the graph above.
a---b    b  d
    |    |\ |
    |    | \|    a---b---c---d
d---c    a  c

This means that in the definition of $P_3$ as a graph with $4$ vertices and $3$ edges "arranged in a linear fashion", the word "arranged" does not refer to how the vertices are drawn. Its formal meaning is that there exists a linear ordering of the $4$ vertices such that there is an edge between each vertex and the next vertex in that ordering.
(Not all graphs with $4$ vertices and $3$ edges have this property: for example, we could take a $3$-vertex cycle and an isolated vertex.)

There is a second subtlety. Under this definition, there are different graphs that are all $P_3$, in a sense. We could have vertices $\{a,b,c,d\}$ and edges $\{ab,bc,cd\}$; we could have vertices $\{v_1, v_2, v_3, v_4\}$ and edges $\{v_1 v_2, v_2 v_3, v_3 v_4\}$; we could have vertices $\{a,b,c,d\}$ and edges $\{ad,db,bc\}$ (which still give us a path graph, but with a different ordering of the vertices). All of these graphs are isomorphic graphs but not the same graph.
Some people are fine with $P_3$ referring to any of these graphs. Some people prefer to say that $P_3$ is one specific example (a bit like the standard kilogram) and all other such graphs are "copies of $P_3$". It shouldn't matter, because in graph theory we basically never discuss properties of a graph that are not preserved under graph isomorphism, and we must be very careful if we do.
