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The word path is used in different way in different contexts. But I can't related them with each other. Like

geeksforgeeks ${}^1$Path: It is a trail in which neither vertices nor edges are repeated i.e. if we traverse a graph such that we do not repeat a vertex and nor we repeat an edge.

here Path: A path in a graph is a subgraph of a given graph that is isomorphic to a path graph.

kenneth rosen discrete mathematics Path: Let n be a nonnegative integer and G a directed graph. A path of length n from u to v in G is a sequence of edges $e_1,\cdots,e_n$ of G such that $e_1$ is associated with $(x_0,x_1)$, $e_2$ is associated with $(x_1,x_2)$, and so on, with $e_n$ is associated with $(x_{n-1},x_n)$, where $x_0=u$ and $x_n=v$. When there are no multiple edges in the directed graph, this path is denoted by its vertex sequence $x_0,x_1,x_2,\cdots,x_n$. path of length greater than zero that begins and ends at the same vertex is called a circuit or cycle. A path or circuit is called simple if it does not contain the same edge more than once.

Question $2$: How many paths of length four are there from $a$ to $d$ in the simple graph $G$ in Figure $8?$enter image description here So there are exactly eight paths of length four from a to d according to their solution. But it can't possible without $\color{red}{\text{repeated vertices}}$. Which seem conflicted to the first definition${}^1$. So I really confused which definition should I follow or I have misunderstood something with these defintions. Besides provide me a list something like:

$(1)$ Walk can be repeated anything
$(2)$ Vertex can be repeated Edges not repeated in Trail
$ \vdots $

Thanks for your time. Thanks in advance .

Emon Hossain
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1 Answers1

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See my answer here and Matthew Daly's comment. In this case, I would say (and many others) that the $ij$th entry $a_{ij}$ of $A^n$ is the number of walks of length $a_{ij}$ from $v_i$ to $v_j$, where $A$ is the adjacency matrix (More information in this stack exchange question.)

In this particular text, it looks like path is defined to be as what is otherwise referred to as a walk, and then they use simple to clarify, which is not uncommon.

I'd recommend taking a look at Douglas West's Introduction to Graph Theory if you can, specifically sections $1.1$ and $1.2$. I believe West defines a path graph in section $1.1$ and implicitly uses the "isomorphic to a path graph definition" in section $1.2$.

Hendrix
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  • Sorry for late response @Hendrix Sir. Yes now it make sense. One question Is it possible for a path that have repeated edges also and Can you suggest me a good book for Graph theory for deep understanding. Because at that time I used kenneth rosen discrete mathematics book which seem not so covering and interesting to read. And Yes I will accept your answer after your response Sir. Thanks again – Emon Hossain Oct 28 '19 at 18:18
  • @emonhossain In Rosen's text, it seems a path can have a repeated edge. In my linked answer no. It all depends on how you choose your definitions. Here are some good recommendations. Diestel is nice and available for free online. My personal experience was learning from Miklos Bona's A Walk Through Combinatorics, and now I use Douglas West's book (while not always friendly, it is insightful and has a wealth of enlightening exercises, and statements are precise.) – Hendrix Oct 28 '19 at 18:30
  • Ok Thanks for you recommendation @Hendrix Sir – Emon Hossain Oct 28 '19 at 18:36